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The Theory of Function Spaces with Matrix Weights
By
Svetlana Roudenko
A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Department of Mathematics
2002
OF The Theory of Function Spaces with Matrix Weights
By
Svetlana Roudenko
AN ABSTRACT OF A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Department of Mathematics
2002
Professor Michael Frazier
ABSTRACT
The Theory of Function Spaces with Matrix Weights
By
Svetlana Roudenko
Nazarov, Treil and Volberg defined matrix Ap weights and extended the theory of
weighted norm inequalities on Lp to the case of vector-valued functions. We develop
some aspects of Littlewood-Paley function space theory in the matrix weight setting.
In particular, we introduce matrix-weighted homogeneous Besov spaces Bαqp (W ) and
matrix-weighted sequence Besov spaces bαqp (W ), as well as bαq
p ({AQ}), where the AQ ’s
are reducing operators for W . Under any of three different conditions on the weight
W , we prove the norm equivalences ‖~f ‖Bαqp (W ) ≈ ‖{~sQ}Q‖bαq
p (W ) ≈ ‖{~sQ}Q‖bαqp ({AQ}) ,
where {~sQ}Q is the vector-valued sequence of ϕ-transform coefficients of ~f . In the
process, we note and use an alternate, more explicit characterization of the matrix
Ap class. Furthermore, we introduce a weighted version of almost diagonality and
prove that an almost diagonal matrix is bounded on bαqp (W ) if W is doubling. We
also obtain the boundedness of almost diagonal operators on Bαqp (W ) under any of
the three conditions on W . This leads to the boundedness of convolution and non-
convolution type Calderon-Zygmund operators (CZOs) on Bαqp (W ), in particular,
the Hilbert transform. We apply these results to wavelets to show that the above
norm equivalence holds if the ϕ-transform coefficients are replaced by the wavelet
coefficients. Next we determine the duals of the homogeneous matrix-weighted Besov
spaces Bαqp (W ) and bαq
p (W ). If W is a matrix Ap weight, then the dual of Bαqp (W )
can be identified with B−αq′p′ (W−p′/p) and, similarly, [bαq
p (W )]∗ ≈ b−αq′p′ (W−p′/p).
Moreover, for certain W which may not be in Ap class, the duals of Bαqp (W ) and
bαqp (W ) are determined and expressed in terms of the Besov spaces B−αq′
p′ ({A−1Q }) and
b−αq′p′ ({A−1
Q }), which we define in terms of reducing operators {AQ}Q associated with
W . We also develop the basic theory of these reducing operator Besov spaces. Fi-
nally, we construct inhomogeneous matrix-weighted Besov spaces Bαqp (W ) and show
that results corresponding to those above are true also for the inhomogeneous case.
ACKNOWLEDGMENTS
I would like to express my deep gratitude to my advisor Prof. Michael Frazier
for introducing the subject matter to me, for his encouragment and belief in me, for
the uncountable number of hours spent sharing his knowledge and discussing various
ideas, and for many useful comments and suggestions while examining my work. I
would also like to express my thanks to Prof.Alexander Volberg for helpful suggestions
and for sharing with me his enthusiasm for and appreciation of mathematics; to Prof.
Peter Yuditskii for fruitful discussions, his availability to help me and his respect; and
to Prof. Fedor Nazarov for conveying his sense of mathematical insight. I thank the
other members of my thesis committee: Prof. William Sledd and Prof. Joel Shapiro.
I would also like to thank Prof. Clifford Weil for introducing me to graduate analysis
and teaching me how to write it well and rigorously.
I would like to say a special thank you to D. Selahi Durusoy for motivation, help
and late night fruit snacks; to Dr. Mark McCormick for his valuable advice at the
beginning of my professional career; to Leo Larsson for helpful discussions as well as
suggestions in technical writing. I want to thank Rebecca Grill and Amy Himes for
their tremendous support throughout my graduate studies. I would also like to thank
Dr. Burak Ozbagci for his support and professional advice. Last but not least, I want
to thank Dr. HsingChi Wang and Dr. David Gebhard for their moral support, time
and help throughout various stages of my doctorate.
iv
TABLE OF CONTENTS
1 Introduction 11.1 History and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Overview of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Notation and Definitions 16
3 Matrix Weights 183.1 The Ap metric and reducing operators . . . . . . . . . . . . . . . . . . . 183.2 Matrix Ap condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3 Two properties of operators . . . . . . . . . . . . . . . . . . . . . . . . . 213.4 The class Bp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.5 δ -doubling measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.6 δ -layers of the Bp class . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.7 Doubling measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.8 Bp implies the doubling property . . . . . . . . . . . . . . . . . . . . . . 293.9 An alternative characterization of the matrix Ap class . . . . . . . . . . 30
4 Boundedness of the ϕ-transform and Its Inverse on Matrix-WeightedBesov Spaces 33
4.1 Boundedness of the inverse ϕ-transform . . . . . . . . . . . . . . . . . . 334.2 Decompositions of an exponential type function . . . . . . . . . . . . . . 404.3 Boundedness of the ϕ-transform . . . . . . . . . . . . . . . . . . . . . . 414.4 Connection with reducing operators . . . . . . . . . . . . . . . . . . . . . 50
5 Calderon-Zygmund Operators on Matrix-Weighted Besov Spaces 535.1 Almost diagonal operators . . . . . . . . . . . . . . . . . . . . . . . . . . 535.2 Calderon-Zygmund operators . . . . . . . . . . . . . . . . . . . . . . . . 63
6 Application to Wavelets 74
7 Duality 777.1 General facts on duality . . . . . . . . . . . . . . . . . . . . . . . . . . . 777.2 Duality of sequence Besov spaces . . . . . . . . . . . . . . . . . . . . . . 777.3 Equivalence of sequence and discrete averaging Besov spaces . . . . . . . 837.4 Properties of averaging Lp spaces . . . . . . . . . . . . . . . . . . . . . . 877.5 Convolution estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
v
7.6 Duality of continuous Besov spaces . . . . . . . . . . . . . . . . . . . . . 957.7 Application of Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
8 Inhomogeneous Besov Spaces 1028.1 Norm equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1028.2 Almost diagonality and Calderon-Zygmund Operators . . . . . . . . . . . 1068.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
9 Weighted Triebel-Lizorkin Spaces 1119.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1119.2 Equivalence of fαq
p (w) and fαqp ({wQ}) . . . . . . . . . . . . . . . . . . . 112
10 Open Questions 116
A Density and convergence 119
BIBLIOGRAPHY 124
vi
CHAPTER 1
Introduction
1.1 History and Motivation
Littlewood-Paley theory gives a unified perspective to the theory of function spaces.
Well-known spaces such as Lebesgue, Hardy, Sobolev, Lipschitz spaces, etc. are
special cases of either Besov spaces Bαqp (homogeneous), Bαq
p (inhomogeneous) or
Triebel-Lizorkin spaces Fαqp (homogeneous), Fαq
p (inhomogeneous) (e.g., see [T]).
These spaces are closely related to their discrete analogues: the sequence Besov spaces
bαqp , bαq
p and sequence Triebel-Lizorkin spaces fαqp , fαq
p ([FJ1], [FJW]). Among other
things, Littlewood-Paley theory provides alternate methods for studying singular in-
tegrals. The Hilbert transform, the classical example of a singular integral operator,
led to the extensive modern theory of Calderon-Zygmund operators, mostly studied
on the Lebesgue Lp spaces.
Motivated by the fundamental result of M. Riesz in the 1920s that the Hilbert
transform preserves Lp for 1 < p < ∞ , Hunt, Muckenhoupt and Wheeden showed
that the famous Ap condition on a weight w is the necessary and sufficient con-
dition for the Hilbert transform to be bounded on Lp(w) (1973, [HMW]). More
recent developments deal with matrix-weighted spaces where scalar methods simply
1
could not be applied. In 1996 Treil and Volberg obtained the analogue of the Hunt-
Muckenhoupt-Wheeden condition for the matrix case when p = 2 ([TV1]). Soon
afterwards, Nazarov and Treil introduced in [NT] a new “Bellman function” method
to extend the theory to 1 < p < ∞ . In 1997 Volberg presented a different solution to
the matrix weighted Lp boundedness of the Hilbert transform via techniques related
to classical Littlewood-Paley theory ([V]).
The purpose of this dissertation is to extend some aspects of Littlewood-Paley
function space theory, previously obtained with no weights and partially for scalar
weights, to the matrix weight setting.
1.2 Overview of the results
We define a new generalized function space: the vector-valued homogeneous Besov
space Bαqp (W ) with matrix weight W . Let M be the cone of nonnegative definite
operators on a Hilbert space H of dimension m (consider H = Cm or Rm ), i.e., for
M ∈ M we have (Mx, x)H ≥ 0 for all x ∈ H . By definition, a matrix weight W is
an a.e. invertible map W : Rn →M . For a measurable ~g = (g1, ..., gm)T : Rn → H ,
let ‖~g‖Lp(W ) =
(∫
Rn
‖W 1/p(t)~g(t)‖pH dt
)1/p
. If the previous norm is finite, then
~g ∈ Lp(W ). We say that a function ϕ ∈ S(Rn) belongs to the class A of admissible
kernels if supp ϕ ⊆ {ξ ∈ Rn : 12≤ |ξ| ≤ 2} and |ϕ(ξ)| ≥ c > 0 if 3
5≤ |ξ| ≤ 5
3. Set
ϕν(x) = 2νnϕ(2νx) for ν ∈ Z .
Definition 1.1 (Matrix-weighted Besov space Bαqp (W )) For α ∈ R, 1 ≤ p <
∞, 0 < q ≤ ∞, ϕ ∈ A and W a matrix weight, the Besov space Bαqp (W ) is the
collection of all vector-valued distributions ~f = (f1, ..., fm)T with fi ∈ S ′/P(Rn), 1 ≤
2
i ≤ m (the space of tempered distributions modulo polynomials) such that
‖~f ‖Bαqp (W ) =
∥∥∥{
2να‖ϕν ∗ ~f ‖Lp(W )
}ν
∥∥∥lq
=∥∥∥{‖W 1/p · (ϕν ∗ ~f )‖Lp
}ν
∥∥∥lαq
< ∞,
where ϕν ∗ ~f = (ϕν ∗ f1, ..., ϕν ∗ fm)T .
The case p = ∞ is not of interest to us, since Bαq∞ (W ) = Bαq
∞ because of the
fact that L∞(W ) = L∞ . Since ϕ is directly involved in the definition of Bαqp (W ),
there seems to be a dependence on the choice of ϕ : Bαqp (W ) = Bαq
p (W,ϕ). Under
appropriate conditions on W , Theorem 1.8 below shows that this is not the case.
The space Bαqp (W ) is complete, as is discussed at the end of Section 4.4.
We also introduce the corresponding weighted sequence (discrete) Besov space
bαqp (W ):
Definition 1.2 (Matrix-weighted sequence Besov space bαqp (W )) For α ∈
R, 1 ≤ p < ∞, 0 < q ≤ ∞ and W a matrix weight, the space bαqp (W ) consists of
all vector-valued sequences ~s = {~sQ}Q , where ~sQ =(s(1)Q , ..., s
(m)Q
)T
, enumerated by
the dyadic cubes Q contained in Rn , such that
‖{~sQ}Q‖bαqp (W ) =
∥∥∥∥∥∥∥
2να
∥∥∥∥∥∥∑
l(Q)=2−ν
|Q|− 12~sQχQ
∥∥∥∥∥∥Lp(W )
ν
∥∥∥∥∥∥∥lq
=
∥∥∥∥∥∥∥
∥∥∥∥∥∥∑
l(Q)=2−ν
|Q|− 12
(‖W 1/p(t)~sQ‖H)χQ(t)
∥∥∥∥∥∥Lp(dt)
ν
∥∥∥∥∥∥∥lαq
< ∞,
where |Q| is the Lebesgue measure of Q and l(Q) is the side length of Q.
For ν ∈ Z and k ∈ Zn , let Qνk be the dyadic cube {(x1, ..., xn) ∈ Rn : ki ≤2νxi < ki + 1, i = 1, ..., n} and xQ = 2−νk is the lower left corner of Qνk . Set
ϕQ(x) = |Q|−1/2ϕ(2νx − k) = |Q|1/2ϕν(x − xQ) for Q = Qνk . For each ~f with
fi ∈ S ′(Rn) we define the ϕ-transform Sϕ as the map taking ~f to the vector-valued
3
sequence Sϕ(~f ) ={⟨
~f, ϕQ
⟩}Q
={
(〈f1, ϕQ〉 , ..., 〈fm, ϕQ〉)T}
Qfor Q dyadic. We
call ~sQ(~f ) :=⟨
~f, ϕQ
⟩the ϕ-transform coefficients of ~f .
The next question is motivated by the following results:
(i) Frazier and Jawerth ([FJ1], 1985) showed that, in the unweighted scalar case,
‖f‖Bαqp≈ ‖{sQ(f)}Q‖bαq
p,
where {sQ(f)}Q are the ϕ-transform coefficients. A similar equivalence holds if
{sQ(f)}Q are the wavelet coefficients {〈f, ψQ〉}Q of f with ψQ being smooth,
say, Meyer’s wavelets (see [M2]).
(ii) Nazarov, Treil and Volberg ([NT], 1996, [V], 1997) obtained
∥∥∥~f∥∥∥
Lp(W )≈
∥∥∥{⟨
~f, hQ
⟩}∥∥∥f02
p (W )if W ∈ Ap, (1.1)
where {hQ}Q is the Haar system and f 02p (W ) is the coefficient (sequence Triebel-
Lizorkin) space for Lp(W ). A particular case of (1.1), when m = 1 and w is a
scalar weight, is
‖f‖B022 (w) = ‖f ‖L2(w) ≈ ‖{〈f, hQ〉}‖f02
2 (w) = ‖{〈f, hQ〉}‖b022 (w),
where the first equality and the second equivalence hold if w ∈ A2 .
For our purposes we will use a condition on W that is equivalent to the matrix
Ap condition of [NT] (for the proof, refer to Section 3.2):
Lemma 1.3 Let W be a matrix weight, 1 < p < ∞, and let p′ be the conjugate of
p (1/p + 1/p′ = 1). Then
∫
B
(∫
B
∥∥W 1/p(x)W−1/p(t)∥∥p′ dt
|B|)p/p′
dx
|B| ≤ cp,n for every ball B ⊆ Rn
(1.2)
if and only if W ∈ Ap .
4
In (1.2), ‖W 1/p(x)W−1/p(t)‖ refers to the matrix (operator) norm.
The advantage of condition (1.2) is that it allows us to understand the Ap condition
in terms of matrices, avoiding metrics ρ , ρ∗ and their averagings as well as reducing
operators (for definitions and details refer to Section 3.2).
Our first result is the norm equivalence between the continuous matrix-weighted
Besov space Bαqp (W ) and the discrete matrix-weighted Besov space bαq
p (W ) under
the Ap condition:
Theorem 1.4 Let α ∈ R, 0 < q ≤ ∞, 1 < p < ∞ and W ∈ Ap . Then
∥∥∥~f∥∥∥
Bαqp (W )
≈∥∥∥∥{~sQ
(~f
)}Q
∥∥∥∥bαqp (W )
. (1.3)
In some cases, the Ap requirement on W can be relaxed. Recall that a scalar
measure µ is called doubling if there exists c > 0 such that for any δ > 0 and any
z ∈ Rn ,
µ(B2δ(z)) ≤ c µ(Bδ(z)), (1.4)
where Bδ(z) = {x ∈ Rn : |z − x| < δ} .
Definition 1.5 (Doubling matrix) A matrix weight W is called a doubling matrix
(of order p, 1 ≤ p < ∞), if there exists a constant c = cp,n such that for any y ∈ H ,
any δ > 0 and any z ∈ Rn ,
∫
B2δ(z)
‖W 1/p(t) y‖pH dt ≤ c
∫
Bδ(z)
‖W 1/p(t) y‖pH dt, (1.5)
i.e., the scalar measure wy(t) = ‖W 1/p(t) y‖pH is uniformly doubling and not identi-
cally zero (a.e.). If c = 2β is the smallest constant for which (1.5) holds, then β is
called the doubling exponent of W .
5
It is known that if W ∈ Ap , then wy is a scalar Ap weight for any y ∈ H and the
Ap constant is independent of y (for example, see [V]). This, in turn, implies that
wy is a scalar doubling measure (e.g., see [St2]) and the doubling constant is also
independent of y . Using decomposition techniques, we prove the equivalence (1.3)
under the doubling assumption on W with the restriction that p is large, and with
no restriction on p in the case when W is a diagonal matrix:
Theorem 1.6 Let α ∈ R, 0 < q ≤ ∞, 1 ≤ p < ∞, and let W be a doubling matrix
of order p with doubling exponent β . Suppose p > β . Then the norm equivalence
(1.3) holds. If W is diagonal, then (1.3) holds for all 1 ≤ p < ∞.
The case of a scalar weight is a particular case of the diagonal matrix weight case,
and thus, the equivalence (1.3) holds just under the doubling condition. This fact is
essentially known (see [FJ2] for the case of Fαqp ); it is proved here for purposes of
comparison and generalization to the diagonal matrix case.
Remark 1.7 One of the directions of the norm equivalence uses only the doubling
property of W with no restrictions (see Corollary 4.6), but the other direction requires
the stated assumptions on W (see Theorem 4.15). Furthermore, the first direction is
obtained from a more general norm estimate involving families of “smooth molecules”
(see Theorem 4.2).
Summarizing Theorems 1.4 and 1.6, the norm equivalence (1.3) holds under any
of the following conditions:
(A1) W ∈ Ap with 1 < p < ∞ ,
(A2) W is a doubling matrix of order p , 1 ≤ p < ∞ , with p > β , where β is the
doubling exponent of W ,
6
(A3) W is a diagonal doubling matrix of order p with 1 ≤ p < ∞ .
Now we will state the independence of the space Bαqp (W,ϕ) from ϕ :
Theorem 1.8 Let ~f ∈ Bαqp (W,ϕ(1)), ϕ(1) ∈ A, α ∈ R, 0 < q ≤ ∞, 1 ≤ p < ∞,
and suppose any of (A1)-(A3) hold. Then for any ϕ(2) ∈ A,
∥∥∥~f∥∥∥
Bαqp (W,ϕ(1))
≈∥∥∥~f
∥∥∥Bαq
p (W,ϕ(2)).
If we use the language of reducing operators (see [V] or Section 3.2), we extend the
norm equivalence (1.3) to a different sequence space, namely bαqp ({AQ}). For each
dyadic cube Q , consider a reducing operator AQ corresponding to the Lp average
over Q of the norm ‖W 1/p · ‖H , i.e.,
‖AQ~u‖H ≈(
1
|Q|∫
Q
‖W 1/p(t)~u‖pH dt
)1/p
for all vector-valued sequences ~u . Note that the assumption that W is a.e. invertible
guarantees that each AQ is invertible. Define the sequence space bαqp ({AQ}) for
α ∈ R, 1 ≤ p < ∞, 0 < q ≤ ∞ as the space containing all vector-valued sequences
{~sQ}Q with
‖{~sQ}Q‖bαqp ({AQ}) =
∥∥∥∥∥∥∥
2να
∥∥∥∥∥∥∑
l(Q)=2−ν
|Q|− 12 (‖AQ~sQ‖H) χQ
∥∥∥∥∥∥Lp(dt)
ν
∥∥∥∥∥∥∥lq
< ∞.
Theorem 1.9 Let α ∈ R, 0 < q ≤ ∞, 1 ≤ p < ∞. Suppose W satisfies any of
(A1)-(A3). Then
∥∥∥~f∥∥∥
Bαqp (W )
≈∥∥∥∥{~sQ
(~f)}
Q
∥∥∥∥bαqp ({AQ})
. (1.6)
In Chapter 5 we study operators on Bαqp (W ) by considering corresponding opera-
tors on bαqp (W ). In [FJW] it was shown that almost diagonal operators are bounded
on bαqp and, thus, on Bαq
p . In Section 5.1 we define a class of almost diagonal matrices
adαqp (β) for the weighted case and show the boundedness of these matrices on bαq
p (W )
if W is a doubling matrix weight:
7
Theorem 1.10 Let α ∈ R, 0 < q ≤ ∞, 1 ≤ p < ∞, and let W be a doubling matrix
of order p with doubling exponent β . Consider A ∈ adαqp (β). Then A : bαq
p (W ) −→bαqp (W ) is bounded.
We say that a continuous linear operator T : S → S ′ is almost diagonal, T ∈ADαq
p (β) , if for some pair of mutually admissible kernels (ϕ, ψ) (see (2.1), Section
2) the matrix (〈TψP , ϕQ〉QP )Q,P dyadic ∈ adαqp (β) (see Section 5.1). Combining the
boundedness of an almost diagonal matrix with the norm equivalence, we obtain the
boundedness of an almost diagonal operator on Bαqp (W ) under any of (A1)-(A3):
Corollary 1.11 Let T ∈ ADαqp (β), α ∈ R, 0 < q < ∞, 1 ≤ p < ∞. Then T is a
bounded operator on Bαqp (W ) if W satisfies any of (A1)-(A3).
In Section 5.2 we consider classical convolution and generalized non-convolution
Calderon-Zygmund operators (CZOs). The following criterion is used: if an operator
T maps “smooth atoms” into “smooth molecules” (see Sections 4.1 and 5.2 for defini-
tions), then T is almost diagonal (Lemma 5.13) and, therefore, bounded on Bαqp (W ).
To show this property for a CZO, the definition of a “smooth molecule” is modified
in order to compensate for the growth of the weight W (note the dependence of the
decay rate of the molecule on the doubling exponent β ), and, thus, more smoothness
of a CZO kernel is required (see Theorems 5.25 and 5.19). In particular, for example,
we obtain the boundedness of the Hilbert transform (when the underlying dimension
is n = 1) and the Riesz transforms (n ≥ 2) on Bαqp (W ) under any of the conditions
(A1)-(A3).
In Chapter 6 we apply the previous results to Meyer’s wavelets and Daubechies’
DN wavelets with N sufficiently large, to show that, instead of the ϕ-transform
coefficients, one can use the wavelet coefficients for the norm equivalence:
8
Theorem 1.12 Let α ∈ R, 0 < q ≤ ∞, 1 ≤ p < ∞ and let W satisfy any of
(A1)-(A3), then∥∥∥~f
∥∥∥Bαq
p (W )≈
∥∥∥∥{~sQ
(~f)}
Q
∥∥∥∥bαqp (W )
,
where{~sQ
(~f)}
Qare the wavelet coefficients of ~f .
The next goal (Chapter 7) is to determine the duals of the Besov function spaces
Bαqp (W ) and the corresponding sequence spaces bαq
p (W ) for α ∈ R , 0 < q < ∞ and
1 < p < ∞ .
To understand what properties of W are needed to identify dual spaces, we will
heavily use the technique of reducing operators (for definitions refer to Section 3.2
or [V]). In fact, instead of dealing with matrix weights, we consider a sequence of
matrices enumerated by dyadic cubes and establish properties of Besov spaces with
such sequences of matrix weights. Then, given a matrix W , its reducing operators
constitute such a sequence.
Denote by D the collection of dyadic cubes in Rn and for each Q ∈ D let AQ be
a positive-definite (thus, self-adjoint) operator on H . Also denote by RSD (reducing
sequences) the collection of all sequences {AQ}Q∈D of positive-definite operators on
H .
In Chapter 7 as a main tool and a useful object by itself, we define the space
Bαqp ({AQ}) with a sequence of discrete weights {AQ}Q :
Definition 1.13 (Averaging matrix-weighted Besov space Bαqp ({AQ})) For
α ∈ R, 1 ≤ p ≤ ∞, 0 < q ≤ ∞, {AQ}Q ∈ RSD and ϕ ∈ A, the Besov space
Bαqp ({AQ}) is the collection of all vector-valued distributions ~f = (f1, ..., fm)T with
fi ∈ S ′/P(Rn), 1 ≤ i ≤ m such that
∥∥∥~f∥∥∥
Bαqp ({AQ})
=
∥∥∥∥∥∥
2να
∥∥∥∥∥∥∑
l(Q)=2−ν
∥∥∥AQ · (ϕν ∗ ~f )∥∥∥H
χQ
∥∥∥∥∥∥Lp
ν
∥∥∥∥∥∥lq
< ∞.
9
This space is well-defined (i.e., independent of ϕ ∈ A), see Corollary 7.10, if {AQ}Q
is a doubling matrix sequence defined as follows.
Definition 1.14 (Doubling sequence) We say {AQ}Q ∈ RSD is a (dyadic) dou-
bling sequence (of order p, 1 ≤ p < ∞), if there exists β ≥ n and c ≥ 1 such that
for all P, Q dyadic
‖AQA−1P ‖p ≤ c
|P ||Q| max
(1,
[l(Q)
l(P )
]β)(
1 +dist (P, Q)
max(l(P ), l(Q))
)β
. (1.7)
Observe that if (1.7) holds for some p , then it holds for 1 ≤ q < p , since the right-hand
side is ≥ 1.
Our main result of this chapter identifies the dual space of Bαqp (W ). For W ∈ Ap
the result can be expressed in terms of matrix weights. However, even for W /∈ Ap but
satisfying (A2) or (A3), we are able to characterize[Bαq
p (W )]∗
in terms of reducing
operators. Set 1p+ 1
p′ = 1 if 1 < p < ∞ and p′ = ∞ if p = 1; 1q+ 1
q′ = 1 if 1 < q < ∞and q′ = ∞ if 0 < q ≤ 1. It is important to emphasize our convention for the duality
pairing. In what follows, we say that a function space Y is a dual of a function space
X , X∗ ≈ Y , in the sense that each y ∈ Y defines an element ly of X∗ via the pairing
ly(x) = (x, y) =∫
Rn 〈x(t), y(t)〉 dt and every element of X∗ is of the kind ly for some
y ∈ Y with ‖ly‖ ≈ ‖y‖Y . (For example, [Lp(W )]∗ ≈ Lp′(W−p′/p), 1 < p < ∞ , with
the pairing (~f,~g ) =∫
Rn
⟨~f(t), ~g(t)
⟩dt ; refer to Section 7.2 for more details.)
Theorem 1.15 Let α ∈ R, 1 ≤ p < ∞, 0 < q < ∞ and let {AQ}Q be reducing
operators of a matrix weight W .
If W ∈ Ap, 1 < p < ∞, then[Bαq
p (W )]∗≈ B−αq′
p′ (W−p′/p). (1.8)
If W satisfies any of (A1)-(A3), then[Bαq
p (W )]∗≈ B−αq′
p′ ({A−1Q }).
(1.9)
10
(For the proof refer to Section 7.4.)
Next we identify the dual space of the sequence (discrete) Besov space bαqp (W ).
Recall that the connection between bαqp (W ) and Bαq
p (W ) is that ~f ∈ Bαqp (W ) if
and only if the appropriate wavelet coefficient sequence of ~f belongs to bαqp (W ).
Analogously to Bαqp ({AQ}) we introduce bαq
p ({AQ}).
Definition 1.16 (Averaging matrix-weighted discrete Besov space
bαqp ({AQ}).) For α ∈ R, 1 ≤ p ≤ ∞, 0 < q ≤ ∞ and {AQ}Q ∈ RSD , the
space bαqp ({AQ}) consists of all vector-valued sequences {~sQ}Q∈D such that
‖{~sQ}Q‖bαqp ({AQ}) =
∥∥∥∥∥∥∥
2να
∥∥∥∥∥∥∑
l(Q)=2−ν
|Q|− 12 (‖AQ~sQ‖H) χQ(t)
∥∥∥∥∥∥Lp(dt)
ν
∥∥∥∥∥∥∥lq
=∥∥∥{AQ~sQ}Q
∥∥∥bαqp
< ∞.
If {AQ}Q is a sequence of reducing operators for a matrix weight W , then the
norm equivalence
bαqp (W ) ≈ bαq
p ({AQ}) (1.10)
holds for any matrix weight W , α ∈ R , 1 ≤ p < ∞ and 0 < q ≤ ∞ by Lemma 4.18.
Theorem 1.17 Let α ∈ R, 1 ≤ p < ∞, 0 < q < ∞ and let {AQ}Q be reducing
operators of a matrix weight W . Then
[bαqp (W )
]∗≈ b−αq′
p′ ({A−1Q }). (1.11)
Moreover, if W ∈ Ap , 1 < p < ∞, then
[bαqp (W )
]∗≈ b−αq′
p′ (W−p′/p). (1.12)
11
The chapter 7 is organized as follows. In Section 7.2 we discuss the discrete Besov
space bαqp (W ). We use a “one at a time reduction” approach meaning we reduce the
space bαqp (W ) in the following order:
bαqp (W ) −→ bαq
p ({AQ}) −→ bαqp (Rm) −→ bαq
p (R1),
where the last two spaces are unweighted vector-valued and scalar-valued discrete
Besov spaces, and then identify the duals in the opposite order. A similar approach
is used for Bαqp (W ).
The fact that each AQ is constant on each dyadic cube Q allows us establish
[bαqp ({AQ})
]∗≈ b−αq′
p′ ({A−1Q }) (1.13)
for any {AQ}Q ∈ RSD , α ∈ R , 0 < q < ∞ , 1 ≤ p < ∞ . If AQ ’s are generated by a
matrix weight W , then combining (1.10) and (1.13), we get (1.11) of Theorem 1.17.
In order to connect b−αq′p′ ({A−1
Q }) with b−αq′p′ ({A#
Q}) ≈ b−αq′p′ (W−p′/p) the matrix
Ap condition is needed, though only for one direction of the embedding; the other
direction is automatic. Thus, the following chain of the embeddings holds for bαqp (W ):
[bαqp (W )
]∗ any W≈[bαqp ({AQ})
]∗≈ b−αq′
p′ ({A−1Q }) Ap≈ b−αq′
p′ ({A#Q})
anyW≈ b−αq′p′ (W−p′/p). (1.14)
This completes the proof of Theorem 1.17.
In Section 7.3 we prove the norm equivalence between Bαqp ({AQ}) and bαq
p ({AQ})for any doubling sequence {AQ}Q . Note that if AQ ’s are generated by a matrix weight
W , then all that is required from the weight is the doubling condition. Compare this
with (A1)-(A3) conditions for the norm equivalence between the original matrix-
weighted spaces.
12
Theorem 1.18 Let α ∈ R, 0 < q ≤ ∞, 1 ≤ p < ∞ and {AQ}Q be a doubling
sequence (of order p). Then
∥∥∥~f∥∥∥
Bαqp ({AQ})
≈∥∥∥∥{~sQ
(~f)}
Q
∥∥∥∥bαqp ({AQ})
.
In Section 7.4 we establish the correspondence between the continuous Besov
spaces Bαqp (W ) and Bαq
p ({AQ}).
Lemma 1.19 Let α ∈ R, 0 < q ≤ ∞ and 1 ≤ p < ∞. If W satisfies any of
(A1)-(A3) and {AQ}Q is a sequence of reducing operators generated by W , then
Bαqp (W ) ≈ Bαq
p ({AQ}).
For one direction of the above equivalence it suffices to have W doubling.
In Section 7.6 it is shown that if {AQ}Q is a doubling sequence of order p , 1 ≤p < ∞ , then
[Bαq
p ({AQ})]∗≈ B−αq′
p′ ({A−1Q }). (1.15)
Using the above duality and equivalence, we get the following chain:
[Bαqp (W )]∗
(1)≈ [Bαqp ({AQ})]∗ ≈ B−αq′
p′ ({A−1Q }) Ap≈ B−αq′
p′ ({A#Q})
(4)≈ B−αq′p′ (W−p′/p), (1.16)
where the equivalences (1) and (4) hold if W and W−p′/p , respectively, satisfy any
of (A1)-(A3). The third equivalence holds under the Ap condition, however, the Ap
condition is needed only for one direction of the embedding. This proves Theorem
1.15.
So far we have dealt only with homogeneous spaces. However, for a number of
applications it is necessary to consider the inhomogeneous distribution spaces (e.g.,
localized Hardy spaces Hploc = F 02
p , 0 < p < ∞ , in particular, H2loc = B02
2 , see [Go]).
13
In Chapter 8 we “transfer” the theory developed up until now to the inhomogeneous
Besov spaces. The main difference is that instead of considering all dyadic cubes,
we consider only the ones with side length l(Q) ≤ 1, and the properties of func-
tions corresponding to l(Q) = 1 are slightly changed. Modifying the definitions of
the ϕ-transform and smooth molecules, we show that all the statements from the
homogeneous case are essentially the same for the inhomogeneous spaces.
In Chapter 9 we study another class of function spaces - scalar weighted Triebel-
Lizorkin spaces. As a starting point of this part we establish the norm equivalence
between the scalar weighted Triebel-Lizorkin space Fαqp (w) and the averaging scalar
weighted sequence Triebel-Lizorkin space fαqp ({wQ}) (see definitions below) if w ∈
A∞ (see Chapter 3).
Definition 1.20 (scalar-weighted Triebel-Lizorkin space Fαqp (w)) For α ∈
R, 0 < p < ∞, 0 < q ≤ ∞, ϕ ∈ A and w a scalar weight, the Triebel-Lizorkin space
Fαqp (w) is the collection of all distributions f ∈ S ′/P(Rn) such that
‖f‖F αqp (w) =
∥∥∥∥∥∥
(∑
ν∈Z(2να|ϕν ∗ f |)q
)1/q∥∥∥∥∥∥
Lp(w)
< ∞,
where the lq -norm is replaced by the supremum on ν if q = ∞.
This space is well-defined if w is a doubling measure (see [FJ2]).
Definition 1.21 (scalar weighted sequence Triebel-Lizorkin space fαqp (w))
For α ∈ R, 0 < p < ∞, 0 < q ≤ ∞ and w a scalar weight, the Triebel-Lizorkin
space fαqp (w) is the collection of all sequences {sQ}Q∈D such that
‖{sQ}Q‖fαqp (w) =
∥∥∥∥∥∥
(∑Q∈D
(|Q|−α
n− 1
2 sQχQ
)q)1/q
∥∥∥∥∥∥Lp(w)
< ∞,
where the lq -norm is again replaced by the supremum on ν if q = ∞.
14
Definition 1.22 (averaging scalar weighted sequence Triebel-Lizorkin
space fαqp ({wQ}).) For α ∈ R, 0 < p < ∞, 0 < q ≤ ∞ and {wQ}Q∈D a sequence
of non-negative numbers, the Triebel-Lizorkin space fαqp ({wQ}) is the collection of all
sequences {sQ}Q∈D such that
‖{sQ}Q‖fαqp ({wQ}) = ‖{w1/p
Q sQ}Q‖fαqp
=
∥∥∥∥∥∥
(∑Q∈D
(|Q|−α
n− 1
2 w1/pQ sQχQ
)q)1/q
∥∥∥∥∥∥Lp
< ∞,
where the lq -norm is again replaced by the supremum on ν if q = ∞.
Appendix contains several proofs on convergence and density.
15
CHAPTER 2
Notation and Definitions
Let z ∈ Rn . Recall that B(z, δ) = {x ∈ Rn : |z− x| < δ} ≡ Bδ(z). If the center z of
the ball is not essential, we will write Bδ for simplicity. In further notation, < V >B
means the average of V over the set B :1
|B|∫
B
V (t) dt . Denote Wν(t) = W (2−νt)
for ν ∈ Z .
For each admissible ϕ ∈ A , there exists ψ ∈ A (see e.g. [FJW, p.54]) such that
∑
ν∈Zϕ(2νξ) · ψ(2νξ) = 1, if ξ 6= 0. (2.1)
A pair (ϕ, ψ) with ϕ, ψ ∈ A and the property (2.1) will be referred to as a pair of
mutually admissible kernels.
Similarly to ϕQ , define ψQ(x) = |Q|−1/2ψ(2νx − k) for Q = Qνk . The inverse
ϕ-transform Tψ is the map taking a sequence s = {sQ}Q to Tψs =∑
Q sQψQ . In the
vector case, Tψ~s =∑
Q ~sQψQ , where ~sQψQ = (s(1)Q ψQ, ..., s
(m)Q ψQ)T . The ϕ-transform
decomposition (see [FJ2] for more details) states that for all f ∈ S ′/P ,
f =∑
Q
〈f, ϕQ〉ψQ =:∑Q
sQψQ. (2.2)
In other words, Tψ ◦ Sϕ is the identity on S ′/P . Observe that if ϕ(x) = ϕ(−x)
(note that ϕ ∈ A), then sQ = 〈f, ϕQ〉 = |Q|1/2(ϕν ∗ f)(2−νk).
16
In order to establish the connection between matrix weighted Besov spaces and
averaging Besov spaces in Chapter 7, we introduce an auxiliary Lp -space:
Definition 2.1 (Averaging space Lp({AQ}, ν)) For ν ∈ Z, 1 ≤ p ≤ ∞ and
{AQ}Q ∈ RSD , the space Lp({AQ}, ν) consists of all vector-valued locally integrable
functions ~f such that
‖~f ‖Lp({AQ},ν) =
∥∥∥∥∥∥∑
l(Q)=2−ν
χQ(t) AQ~f(t)
∥∥∥∥∥∥Lp(dt)
< ∞.
Note that∥∥∥~f
∥∥∥Bαq
p ({AQ})=
∥∥∥∥{
2να∥∥∥ϕν ∗ ~f
∥∥∥Lp({AQ},ν)
}
ν
∥∥∥∥lq
.
To make notation short, define Qν = {Q ∈ D : l(Q) = 2−ν} .
17
CHAPTER 3
Matrix Weights
3.1 The Ap metric and reducing operators
Let t ∈ Rn . Consider the family of norms ρt : H −→ R+ . Then the dual norm ρ∗ is
given by
ρ∗t (x) = supy 6=0
|(x, y)|ρt(y)
.
Following [V] (or [NT], [TV1]), we introduce the norms ρp,B through the averagings
of the metrics ρt over a ball B
ρp,B(x) =
(1
|B|∫
B
[ρt(x)]p dt
)1/p
.
Similarly, for the dual norm
ρ∗p′,B(x) =
(1
|B|∫
B
[ρ∗t (x)]p′dt
)1/p′
.
Definition 3.1 (Ap - metric) The metric ρ is an Ap -metric, 1 < p < ∞, if
ρ∗p′,B ≤ C (ρp,B)∗ for every ball B ⊆ Rn. (3.1)
The condition (3.1) is equivalent to
ρp,B ≤ C (ρ∗p′,B)∗ for every ball B ⊆ Rn,
18
which means that ρ∗ is an Ap′ -metric.
If ρ is a norm on H , then there exists a positive operator A , which is called a
reducing operator of ρ , such that
ρ(x) ≈ ‖Ax‖ for all x ∈ H.
For details we refer the reader to [V]. Let AB be a reducing operator for ρp,B , and
A#B for ρ∗p′,B . Then, in the language of the reducing operators, the condition (3.1) for
the Ap class is
‖A#B AB‖ ≤ C < ∞ for every ball B ⊆ Rn. (3.2)
Proof. Since ρ∗p′,B(x) ≈ ‖A#Bx‖ and (ρp,B)∗(x) = sup
y 6=0
|(x, y)|ρp,B(y)
, (3.3) implies
‖A#Bx‖ ≤ c sup
y 6=0
|(x, y)|‖AB y‖ = c sup
z 6=0
|(x,A−1B z)|
‖z‖ , where z = AB y.
Since A−1B is self-adjoint,
‖A#Bx‖ ≤ c sup
z 6=0
|(A−1B x, z)|‖z‖ = c ‖A−1
B x‖.
With u = A−1B x , we obtain
‖A#B ABu‖ ≤ c ‖u‖, or ‖A#
B AB‖ ≤ c.
¥
Note that the opposite inequality ‖(A#Q AQ)−1‖ ≤ c holds always as a simple
consequence of Holder’s inequality: for any x , y ∈ H we have
|(x, y)| ≤(∫
Q
‖W 1/p(t) x‖p dt
|Q|)1/p (∫
Q
‖W−1/p(t) y‖p′ dt
|Q|)1/p′
≈ ‖AQ x‖ ‖A#Q y‖,
which implies ‖AQ x‖ ≥ c ‖(A#Q)−1 x‖ for any x ∈ H and, thus, the above statement
follows.
19
3.2 Matrix Ap condition
The particular case of norms ρt , we will be interested from now on, is
ρt(x) = ‖W 1/p(t) x‖, x ∈ H, t ∈ Rn.
Then the dual metric ρ∗t is given by
ρ∗t (x) = supy 6=0
|(x, y)|ρt(y)
= ‖W−1/p(t) x‖.
Definition 3.2 (Matrix Ap weight) For 1 < p < ∞, we say that a matrix weight
W is an Ap matrix weight if there exists C < ∞ such that for every ball B ⊆ Rn
ρ∗p′,B ≤ C (ρp,B)∗, (3.3)
where both averaging metrics are generated by W , i.e.,
ρp,B(x) =
(1
|B|∫
B
‖W 1/p(t) x‖p dt
)1/p
and
ρ∗p′,B(x) =
(1
|B|∫
B
‖W−1/p(t) x‖p′ dt
)1/p′
.
Remark 3.3 If p = 2, the condition A2 simplifies significantly:
‖ < W >1/2B < W−1 >
1/2B ‖ ≤ C for every ball B ⊆ Rn. (3.4)
Proof.
[ρp,B(x)]2 =
∫
B
‖W 1/2(t) x‖2 dt
|B| =
∫
B
(W (t) x, x)dt
|B|= (< W >B x, x) = ‖ < W >
1/2B x‖2.
This means that a reducing operator AB can be chosen explicitly as < W >1/2B .
Similarly, ρ∗p′,B(x) = ‖ < W−1 >1/2B x‖ and, thus, A#
B ≈< W−1 >1/2B . Therefore,
(3.4) follows from (3.2). ¥
20
Remark 3.4 If w is a scalar weight, the condition Ap is the celebrated Muckenhoupt
Ap condition:
(∫
B
w(t) dt
)1/p (∫
B
w−p′/p(t) dt
)1/p′
≤ c for every ball B ⊆ Rn. (3.5)
Denote wx(t) = ‖W 1/p(t) x‖p and w∗x(t) = ‖W−1/p(t) x‖p′ . Similarly, w(t) =
‖W 1/p(t)‖p and w∗(t) = ‖W−1/p(t)‖p′ . Sometimes it is more convenient to work
with these families of scalar-valued measures.
3.3 Two properties of operators
Observe the following two useful facts. First, if P and Q are two selfadjoint operators
in a normed space, then
‖PQ‖ = ‖(PQ)∗‖ = ‖Q∗P ∗‖ = ‖QP‖. (3.6)
Thus, the operators can be commuted as long as we deal with norms.
Second, we need the following lemma:
Lemma 3.5 (Norm lemma) If {e1, . . . , em} is any orthonormal basis in a Hilbert
space H , then for any linear operator V : H → H and r > 0,
‖V ‖r ≈(r,m)
m∑i=1
‖V ei‖rH.
Proof. With xi = (x, ei)H , we get ‖V ‖r = sup‖x‖≤1
‖Vm∑
i=1
xiei‖rH
≤ cr sup‖x‖≤1
m∑i=1
|xi|r‖V ei‖rH ≤ cr
m∑i=1
‖V ei‖rH ≤ crm ‖V ‖r . ¥
3.4 The class Bp
Definition 3.6 For 1 < p < ∞ the class Bp is the collection of all matrix weights
W so that for a given fixed 0 < δ < 1 there exists a constant c = cδ,p,n such that for
21
any z ∈ Rn and any ν ∈ Z the following inequality holds
∫
Bδ(z)
(∫
Bδ(z)
∥∥W 1/pν (x)W−1/p
ν (t)∥∥p′ dt
|Bδ|)p/p′
dx
|Bδ| ≤ cδ,p,n, (3.7)
where Wν(t) = W (2−νt).
This condition seems to be dependent on the choice of δ , though it is not the fact.
Proof. By changing variables we write (3.7) as
∫
B2−νδ(2−νz)
(∫
B2−νδ(2−νz)
∥∥W 1/p(x)W−1/p(t)∥∥p′ dt
|B2−νδ|
) pp′
dx
|B2−νδ|≤ cδ,p,n.
(3.8)
Let ε > 0. Then there exists ν0 ∈ Z such that 2−(ν0+1)δ ≤ ε < 2−ν0δ . The following
three simple observations will show that (3.7) is independent of δ .
1. B2−(ν0+1)δ(z) ⊆ Bε(z) ⊆ B2−ν0δ(z),
2. |B2−ν0δ| = 2n|B2−(ν0+1)δ| ,
3.1
2n
∫
B2−(ν0+1)δ
...dt
|B2−(ν0+1)δ|≤
∫
Bε
...dt
|Bε| ≤ 2n
∫
B2−ν0δ
...dt
|B2−ν0δ|.
¥
Hence, the definition of the Bp class is equivalent to the following one:
Definition 3.7 The class Bp , 1 < p < ∞, is the collection of all matrix weights W
so that there exists a constant c = cp,n such that for any z ∈ Rn and any ε > 0 the
inequality (1.2) holds, i.e.,
∫
Bε(z)
(∫
Bε(z)
∥∥W 1/p(x)W−1/p(t)∥∥p′ dt
|Bε|)p/p′
dx
|Bε| ≤ cp,n. (3.9)
Remark 3.8 It is also convenient to write condition (3.9) in terms of metrics ρ and
ρ∗ :
∫
Bε(z)
(∫
Bε(z)
supy 6=0
[ρ∗t (y)
ρ∗x(y)
]p′dt
|Bε|
)p/p′dx
|Bε| ≤ cp,n, (3.10)
22
or
∫
Bε(z)
(∫
Bε(z)
supy 6=0
[ρx(y)
ρt(y)
]p′dt
|Bε|
)p/p′dx
|Bε| ≤ cp,n. (3.11)
Proof. We will show only (3.10), since (3.11) uses the same argument. The
left-hand side of (3.10) is equal to
∫
Bε(z)
(∫
Bε(z)
supy 6=0
‖W−1/p(t)y‖p′
‖W−1/p(x)y‖p′dt
|Bε|)p/p′
dx
|Bε| .
Let u = W−1/p(x) y , then the last expression is
∫
Bε(z)
(∫
Bε(z)
supu6=0
‖W−1/p(t)W 1/p(x)u‖p′
‖u‖p′dt
|Bε|)p/p′
dx
|Bε|
=
∫
Bε(z)
(∫
Bε(z)
‖W−1/p(t)W 1/p(x)‖p′ dt
|Bε|)p/p′
dx
|Bε| ,
which is (3.9), by (3.6). ¥
Remark 3.9 Similarly, (3.7) can be written in terms of metrics ρ and ρ∗ :
∫
Bδ(z)
∫
Bδ(z)
supy 6=0
[ρ∗(2−νt)(y)
ρ∗(2−νx)(y)
]p′dt
|Bδ|
p/p′
dx
|Bδ| ≤ cδ,p,n for any ν ∈ Z,
(3.12)
or
∫
Bδ(z)
(∫
Bδ(z)
supy 6=0
[ρ(2−νx)(y)
ρ(2−νt)(y)
]p′dt
|Bδ|
)p/p′dx
|Bδ| ≤ cδ,p,n for any ν ∈ Z.
(3.13)
Proof. We will show only (3.12), since (3.13) uses the same argument. The
condition (3.12) is equal to
∫
Bδ(z)
(∫
Bδ(z)
supy 6=0
‖W−1/pν (t)y‖p′
‖W−1/pν (x)y‖p′
dt
|Bδ|
)p/p′dx
|Bδ| .
23
Let u = W−1/pν (x) y , then the last expression is
∫
Bδ(z)
(∫
Bδ(z)
supu6=0
‖W−1/pν (t)W
1/pν (x)u‖p′
‖u‖p′dt
|Bδ|
)p/p′dx
|Bδ|
=
∫
Bδ(z)
(∫
Bδ(z)
‖W−1/pν (t)W 1/p
ν (x)‖p′ dt
|Bδ|)p/p′
dx
|Bδ| ,
which is (3.7), by (3.6). ¥
With the help of the Norm Lemma, we observe
< w >B=
∫
B
‖W 1/p(t)‖p dt
|B| ≈∫
B
max1≤i≤m
‖W 1/p(t)ei‖p dt
|B|
≈ max1≤i≤m
∫
B
[ρt(ei)]p dt
|B| = max1≤i≤m
[ρp,B(ei)]p ≈ ‖ρp,B‖p.
The last equivalence can also be viewed in terms of reducing operators
< w∗ >B=
∫
B
‖W 1/p(t)‖p dt
|B| ≈ max1≤i≤m
‖ABei‖p ≈ ‖AB‖p.
Similarly, the dual metrics
∫
B
‖W−1/p(t)‖p′ dt
|B| ≈ max1≤i≤m
‖ρ∗p′,B(ei)‖p′ ≈ ‖ρ∗p′,B‖p′ ≈ ‖A∗B‖p′ .
3.5 δ-doubling measures
First, recall that a scalar measure µ is called doubling, if there exists c > 0 such that
for any δ > 0 and any z ∈ Rn the inequality (1.4) holds, i.e.,
µ(B2δ(z)) ≤ c µ(Bδ(z)).
If the above inequality holds only for a specific δ > 0, then we say µ is δ -doubling.
Definition 3.10 Fix y ∈ H . Then wy(t) = ‖W 1/p(t)y‖p is a scalar valued δ -
doubling measure (of order p), 1 ≤ p < ∞, if there exists δ > 0 and a constant
c = cδ,p,n such that for any z ∈ Rn
wy(B2δ(z)) ≤ cwy(Bδ(z)). (3.14)
24
Remark 3.11 Note that if wy(t) is δ -doubling of order p for any y ∈ H , then
w(t) = ‖W 1/p(t)‖p is also a scalar-valued δ -doubling measure of order p.
Proof. Since (3.14) is true for any ei - an orthonormal basis vector of H , we
havem∑
i=1
∫
B2δ
‖W 1/p(t)ei‖p dt ≤ c
m∑i=1
∫
Bδ
‖W 1/p(t)ei‖p dt.
By the Norm Lemma, this inequality is equivalent to
∫
B2δ
‖W 1/p(t)‖p dt ≤ c
∫
Bδ
‖W 1/p(t)‖p dt.
¥
The reverse of the previous remark is not always true.
3.6 δ-layers of the Bp class
Lemma 3.12 Fix δ > 0. Suppose that the condition (3.7) or, equivalently, (3.12)
is true for ν = −1. Then w∗y(t) = ‖W−1/p(t)y‖p′ is a δ -doubling measure for any
y ∈ H .
Proof. By Holder’s inequality
2−n =|Bδ||B2δ| =
∫
B2δ
χBδ(t)
dt
|B2δ| =
∫
B2δ
χBδ(t)
ρ∗t (y)
ρ∗t (y)
dt
|B2δ|
≤(∫
B2δ
χBδ(t)[ρ∗t (y)]p
′ dt
|B2δ|)1/p′ (∫
B2δ
1
[ρ∗x(y)]pdx
|B2δ|)1/p
=
[w∗
y(Bδ)
|B2δ|]1/p′ (∫
B2δ
1
[ρ∗x(y)]pdx
|B2δ|)1/p
=
[w∗
y(Bδ)
w∗y(B2δ)
]1/p′ (∫
B2δ
[ρ∗t (y)]p′ dt
|B2δ|)1/p′ (∫
B2δ
1
[ρ∗x(y)]pdx
|B2δ|)1/p
.
25
Raising to the pth power both sides of the previous chain and specifying z as a center
of both balls Bδ and B2δ , we get
2−np =
( |Bδ||B2δ|
)p
≤[
w∗y(Bδ(z))
w∗y(B2δ(z))
]p/p′
·∫
B2δ(z)
(∫
B2δ(z)
[ρ∗t (y)
ρ∗x(y)
]p′dt
|B2δ|
)p/p′dx
|B2δ|
=
[w∗
y(Bδ(z))
w∗y(B2δ(z))
]p/p′
·∫
Bδ(z/2)
(∫
Bδ(z/2)
[ρ∗2t(y)
ρ∗2x(y)
]p′dt
|Bδ|
)p/p′dx
|Bδ|
≤ cδ,p,n
[w∗
y(Bδ(z))
w∗y(B2δ(z))
]p/p′
,
by the Bp condition in terms of metrics (3.12) with ν = −1.
Simplifying the last chain, we get
w∗y(B2δ(z)) ≤ (2np′ · c p′/p
δ,p,n) w∗y(Bδ(z)), (3.15)
i.e., w∗y is a δ -doubling measure. ¥
Remark 3.13 Repeating the same argument, it can be shown that wy is also a δ -
doubling measure.
Proof. The proof is similar to the previous one, thougn the splitting of the initial
equality is slightly tricky. So, by Holder’s inequality
2−np =
( |Bδ||B2δ|
)p
=
(∫
B2δ(z)
χBδ(z)(t)dt
|B2δ|)p
=
(∫
B2δ(z)
χBδ(z)(t)ρt(y)
ρt(y)
dt
|B2δ|)p
≤(∫
B2δ(z)
χBδ(z)(t)[ρx(y)]pdx
|B2δ|)(∫
B2δ(z)
1
[ρt(y)]p′dt
|B2δ|)p/p′
=
[wy(Bδ(z))
|B2δ|](∫
B2δ(z)
1
[ρt(y)]p′dt
|B2δ|)p/p′
=
[wy(Bδ(z))
wy(B2δ(z))
](∫
B2δ(z)
[ρx(y)]pdx
|B2δ|)(∫
B2δ(z)
1
[ρt(y)]p′dt
|B2δ|)p/p′
=
[wy(Bδ(z))
wy(B2δ(z))
] ∫
Bδ(z/2)
(∫
Bδ(z/2)
[ρ2x(y)
ρ2t(y)
]p′dt
|Bδ|
)p/p′dx
|Bδ| ≤ cδ,p,n
[wy(Bδ(z))
wy(B2δ(z))
],
26
by the Bp condition in terms of metrics (3.13) with ν = −1.
Simplifying, we get
wy(B2δ(z)) ≤ (2np · cδ,p,n) wy(Bδ(z)), (3.16)
i.e., wy is a δ -doubling measure. ¥
Generalizing the previous lemmas, we get
Corollary 3.14 Fix δ > 0. Then w∗ν,y(t) := ‖W−1/p
ν (t)y‖p′ and wν,y(t) :=
‖W 1/pν (t) y‖p are δ -doubling measures for any y ∈ H , if the condition (3.7) or, equiv-
alently, (3.12) holds for ν − 1.
Proof. Let V (t) = Wν(t), then V−1(t) = Wν−1(t), and so (3.7) holds for V
with ν = −1. Applying previous lemma (3.12) to u∗y(t) := ‖V −1/p(t)y‖p′ , we get
u∗y is δ -doubling, or, u∗y(t) = ‖V −1/p(t)y‖p′ = ‖W−1/pν (t)y‖p′ = w∗
ν,y(t) is δ -doubling.
Analogous proof applies to wν,y . ¥
So each “layer” of the Bp condition implies δ -doubling property of the scalar-
valued measures generated by the matrix weight W . Anticipating further results,
one can predict that the whole Bp class will imply a standard doubling property.
3.7 Doubling measures
Let W be a doubling matrix of order p , i.e., (1.5) holds for any y ∈ H , δ > 0 and
z ∈ Rn . For p = 2 this simplifies to
∫
B2δ
W (t) dt ≤ c
∫
Bδ
W (t) dt (3.17)
for a given δ , where the inequality is understood in the sense of selfadjoint operators.
27
Remark 3.15 Note that ‖W 1/p(t)‖p is independent of p. If wy(t) = ‖W 1/p(t) y‖pH
is doubling of order p for any y ∈ H , then w(t) = ‖W 1/p(t)‖p is also a scalar-valued
doubling measure.
Proof. Fix t ∈ Rn . Then there exist a unitary matrix U and a diagonal matrix
Λ such that W (t) = U Λ U−1 , and so W 1/p(t) = U Λ1/p U−1 . Moreover, since the
norm of a positive diagonal matrix is the largest eigenvalue, say λ0 , ‖W 1/p(t)‖ = λ1/p0
and, hence, ‖W 1/p(t)‖p = λ0 , regardless of what p is.
Now, since (1.5) is true with y = ei - any orthonormal basis vector of H , by the
Norm Lemma we get the second assertion:
∫
B2δ
‖W 1/p(t)‖p dt ≈m∑
i=1
∫
B2δ
‖W 1/p(t)ei‖p dt
≤ c
m∑i=1
∫
Bδ
‖W 1/p(t)ei‖p dt ≈ c
∫
Bδ
‖W 1/p(t)‖p dt.
¥
The doubling property of w(t) = ‖W 1/p(t)‖p is not very helpful if one wants to
understand the nature of W ; it only tells us how large the weight is, not how it is
distributed in different directions. Therefore, we use the definition of doubling matrix
in (1.5), which involves different directions of y ∈ H .
Remark 3.16 In the scalar case, (1.5) gives the standard doubling measure:
∫
B2δ
w(t)|y|p dt ≤ c
∫
Bδ
w(t)|y|p dt,
and if y 6= 0, then w(B2δ) ≤ cw(Bδ). In particular, there is no dependence on p in
the scalar situation.
Similar definitions for doubling weights (of order p′ ) can be analogously given for
the “dual” measure w∗y(t) = ‖W−1/p(t)y‖p′ .
28
Remark 3.17 The doubling property (1.4) is equivalent to
µ(F )
µ(E)≤ c
( |F ||E|
)β/n
, (3.18)
where F is a ball (or a cube) and E ⊆ F is a sub-ball (or a sub-cube) (not any subset
of F ; any subset would be equivalent to the A∞ condition, see the end of Section 3.9,
also [St2]).
Proof. Since E ⊆ F , there exists j ∈ N such that 2jE ≈ F , i.e., l(F ) ≈ 2jl(E).
Since µ is doubling, by (1.4) we haveµ(F )
µ(E)≤ cj ≈ clog2
l(F )l(E) . Noticing that
|F ||E| =
[l(F )
l(E)
]n
, we get (3.18). ¥
In further estimates, it is more convenient to use (3.18) instead of (1.4).
Observe that the doubling exponent of the Lebesgue measure in Rn is β = n ;
moreover, if µ is any nonzero doubling measure in Rn , then β(µ) ≥ n .
It is a trivial fact that if W is a doubling matrix weight (of order p), then a
reducing operator sequence {AQ}Q , generated by W , is a doubling sequence (of
order p). (Recall the definition 1.14.)
3.8 Bp implies the doubling property
Corollary 3.18 Let W ∈ Bp . Then w∗ν,y(t) = ‖W−1/p
ν (t)y‖p′ and wν,y(t) =
‖W 1/pν (t)y‖p are doubling measures for any y ∈ H and any ν ∈ Z.
Proof. First, by the Lemma (3.14) w∗ν,y(t) and wν,y(t) are δ -doubling for any ν .
Second, if W ∈ Bp , then Wν satisfies (3.7) for all ν ∈ Z and a given 0 < δ < 1. But
we know that the Bp class is independent of the choice of δ , which means w∗ν,y(t) and
wν,y(t) are δ -doubling measures for any δ . Therefore, the corollary follows trivially.
¥
29
Lemma 3.19 Let x ∈ H and W ∈ Ap . Then vx(t) := ‖W 1/p(x)W−1/p(t)‖p′ =
‖W−1/p(t)W 1/p(x)‖p′ is a doubling measure, i.e., there exists a constant c such that
for any δ > 0
∫
B2δ
‖W 1/p(x)W−1/p(t)‖p′dt ≤ c
∫
Bδ
‖W 1/p(x)W−1/p(t)‖p′dt. (3.19)
Proof. Applying the Norm Lemma to the operator norm in the left-hand side,
we obtain
vx(t) ≈m∑
i=1
‖W−1/p(t)W 1/p(x)ei‖p′ =m∑
i=1
‖W−1/p(t)yi(x)‖p′ =m∑
i=1
w∗yi(x)(t),
where yi(x) = W 1/p(x)ei . Then
vx(B2δ) ≈m∑
i=1
∫
B2δ
w∗yi(x)(t) dt ≤
m∑i=1
c
∫
Bδ
w∗yi(x)(t) dt ≤ c vx(Bδ),
since w∗y is doubling (W−p′/p ∈ Ap′ ). ¥
3.9 An alternative characterization of the matrix
Ap class
or What is the Bp condition indeed?
Now we are ready to reveal what the class Bp really is, or, in other words, we give
a proof of the equivalence of condition (3.9), or (1.2), to the Ap condition.
Proof of Lemma 1.3. By property (3.6) and the Norm Lemma
∫
B
(∫
B
∥∥W 1/p(x)W−1/p(t)∥∥p′ dt
|B|)p/p′
dx
|B|
=
∫
B
(∫
B
∥∥W−1/p(t)W 1/p(x)∥∥p′ dt
|B|)p/p′
dx
|B|
≈∫
B
(∫
B
m∑i=1
∥∥W−1/p(t)W 1/p(x)ei
∥∥p′ dt
|B|
)p/p′dx
|B|
30
≈m∑
i=1
∫
B
(∫
B
[ρ∗t (W
1/p(x) ei)]p′ dt
|B|)p/p′
dx
|B| =m∑
i=1
∫
B
[ρ∗p′,B(W 1/p(x) ei)
]p dx
|B| .
Now, in terms of the reducing operators, the last expression is equivalent to
m∑i=1
∫
B
∥∥∥A#B(W 1/p(x) ei)
∥∥∥p dx
|B| ≈∫
B
∥∥∥A#B W 1/p(x)
∥∥∥p dx
|B|
≈m∑
i=1
∫
B
∥∥∥W 1/p(x)(A#B ei)
∥∥∥p dx
|B| ≈m∑
i=1
[ρp,B(A#
B ei)]p
≈m∑
i=1
∥∥∥AB(A#B ei)
∥∥∥p
≈∥∥∥AB A#
B
∥∥∥p
.
Therefore, (1.2) is equivalent to∥∥∥A#
B AB
∥∥∥p
≤ c , i.e., the Ap condition. ¥
Thus, the Bp class is nothing else but the matrix Ap class. Therefore, we will
not use the notation Bp anymore, though it was useful to understand what layers
this class consists of (as well as Ap ) and that each layer implies a certain doubling
property.
Remark 3.20 Rephrasing Corollary 3.18, we obtain that Ap implies doubling.
Moreover, if W ∈ Ap , then by (3.16) W is the doubling matrix weight of order p
and the doubling exponent β ≤ np + log2 cp,n = p
(n +
log2 cp,n
p
), where cp,n is the
constant in (1.2).
Also W ∈ Ap implies that the “dual” weight W−p′/p is a doubling matrix of order
p′ with the doubling exponent β′ ≤ p′(
n +log2 cp,n
p
)by using (3.15), where again
cp,n is the constant from (1.2).
Corollary 3.21 (Symmetry of matrix Ap condition) The following state-
ments are equivalent:
(i) W ∈ Ap ;
(ii) W−p′/p ∈ Ap′ ;
31
(iii)
∫
B
(∫
B
∥∥W 1/p(x)W−1/p(t)∥∥p′ dt
|B|)p/p′
dx
|B| ≤ c for every ball B ⊆ Rn ;
(iv)
∫
B
(∫
B
∥∥W 1/p(x)W−1/p(t)∥∥p dx
|B|)p′/p
dt
|B| ≤ c for every ball B ⊆ Rn .
Proof. Recall that ρ ∈ Ap if and only if ρ∗ ∈ Ap′ . In terms of matrix weights,
W ∈ Ap if and only if W−p′/p ∈ Ap′ (note that ρ∗t (x) = ‖(W−p′/p)1/p′(t) x‖). By
Lemma 1.3, the third statement is equivalent to W ∈ Ap , whereas the fourth is
equivalent to W−p′/p ∈ Ap′ . ¥
Observe that the scalar classes Ap are increasing in p , i.e., Ap ⊆ Aq if p ≤ q .
This observation brings us to the definition of the scalar A∞ class.
Definition 3.22 (Scalar A∞ class) Let w ≥ 0. Then A∞ =⋃
1≤p<∞Ap .
Equivalently, w ∈ A∞ if there exists δ > 0 such that given a cube (or a ball) F
and any subset E ⊆ F ,
w(F )
w(E)≤ c
( |F ||E|
)δ
.
(See [St2] for equivalence and other details.)
This property of scalar weights will be used in Chapter 9.
32
CHAPTER 4
Boundedness of the ϕ-transform
and Its Inverse on
Matrix-Weighted Besov Spaces
4.1 Boundedness of the inverse ϕ-transform
Consider Bαqp (W ) with parameters α ∈ R , 0 < q ≤ ∞ , 1 ≤ p < ∞ fixed. For
0 < δ ≤ 1, M > 0 and N ∈ Z define (as in [FJ2]) mQ to be a smooth (δ,M, N)-
molecule for Q ∈ D if:
(M1)
∫xγmQ(x) dx = 0, for |γ| ≤ N ,
(M2) |mQ(x)| ≤ |Q|−1/2
(1 +
|x− xQ|l(Q)
)−max(M,M−α)
,
(M3) |DγmQ(x)| ≤ |Q|−1/2−|γ|/n
(1 +
|x− xQ|l(Q)
)−M
if |γ| ≤ [α],
(M4) |DγmQ(x)−DγmQ(y)| ≤ |Q|− 12− |γ|
n− δ
n |x− y|δ
× sup|z|≤|x−y|
(1 +
|x− z − xQ|l(Q)
)−M
if |γ| = [α].
33
It is understood that (M1) is void if N < 0; and (M3), (M4) are void if α < 0.
Also, [α] stands for the greatest integer ≤ α ; γ is a multi-index γ = (γ1, . . . , γn)
with γi ∈ N ∪ {0} , 1 ≤ i ≤ n , and the standard notation is used.
We say {mQ}Q is a family of smooth molecules for Bαqp (W ) if each mQ is a
(δ,M, N)-molecule with
(M.i) α− [α] < δ ≤ 1,
(M.ii) M > J , where J = βp
+ np′ (if p = 1, then n/p′ := 0 and J = β ),
(M.iii) N = max([J − n− α],−1).
Remark 4.1 Note that, in contrast to the case in [FJ2], there is a dependence of
the family of smooth molecules for Bαqp (W ) on the weight W (more precisely, on the
doubling exponent β ).
Theorem 4.2 Let α ∈ R, 1 ≤ p < ∞, 0 < q ≤ ∞, and let W be a doubling matrix
weight of order p. Suppose {mQ}Q is a family of smooth molecules for Bαqp (W ).
Then
∥∥∥∥∥∑Q
~sQ mQ
∥∥∥∥∥Bαq
p (W )
≤ c ‖{~sQ}Q‖bαqp (W ). (4.1)
The proof uses the following estimates for Q dyadic with l(Q) = 2−µ , µ ∈ Z , and
ϕν , ν ∈ Z , with ϕ ∈ A :
if µ > ν , then for some σ > J − α
|ϕν ∗mQ(x)| ≤ c |Q|−1/2 2−(µ−ν)σ (1 + 2ν |x− xQ|)−M ; (4.2)
if µ ≤ ν , then for some τ > α
|ϕν ∗mQ(x)| ≤ c |Q|−1/2 2−(ν−µ)τ (1 + 2µ|x− xQ|)−M . (4.3)
34
The proofs are entirely elementary, but quite tedious (see [FJ2, Appendix B]).
Note that in the statement of Lemma B.1 in [FJ2], it should say j ≤ k . For (4.2),
for N 6= −1, apply Lemma B.1 with j = ν , k = µ , L = N , R = M , S = M − α ,
g = 2−νn/2 ϕν , h = mQ with l(Q) = 2−µ , x1 = xQ , J − n− α− [J − n− α] < θ ≤ 1.
Letting σ = N + n + θ > J − α , we obtain (4.2). For N = −1, apply Lemma B.2
in [FJ2] with σ = n > J − α to get (4.2). Now for (4.3), for α > 0, apply Lemma
B.1 with k = ν , j = µ , L = [α] , R = M , δ = θ , S = [α] + n + δ , x1 = 0,
g(x) = mQ(x+xQ), h = 2−νn/2 ϕν , and observe that ϕν ∗mQ(x) = 2νn/2 g ∗h(x−xQ)
to get (4.3) with τ = δ + [α] > α . For α < 0, Lemma B.2 in [FJ2] gives (4.3) with
τ = 0 > α .
Lemma 4.3 (Squeeze Lemma) Fix a dyadic cube Q and let w : Rn → R+ be a
scalar doubling measure with the doubling exponent β . If L > β , then for r ≥ l(Q),
∫
Rn
w(x)
(1 +
|x− xQ|r
)−L
dx ≤ cβ
[r
l(Q)
]β ∫
Q
w(x) dx. (4.4)
Proof. Decompose Rn into the annuli Rm :
Rn =∞⋃
m=1
{x : 2m−1r ≤ |x− xQ| < 2mr} ∪ {x : |x− xQ| < r} =:∞⋃
m=0
Rm.
Then the left-hand side of (4.4) is bounded by
∞∑m=1
(1 + 2m−1)−Lw(Rm) + w(R0). (4.5)
Using the doubling property of w , we get
w(Rm) ≤ w(B(xQ, 2mr)) ≤ c
( |B(xQ, 2mr)||R0|
)β/n
w(R0) = c 2mβw(R0).
Thus, (4.5) is bounded by
c
∞∑m=0
2mβ−mL w(R0) ≤ cβ w(R0),
35
since L > β . Note that B(xQ, l(Q)) ⊆ 3Q and so w(B(xQ, l(Q))) ≤ cβ w(Q). If
r > l(Q), then
w(R0) ≤ c
( |R0||B(xQ, l(Q))|
)β/n
w(B(xQ, l(Q))) ≤ cβ
[r
l(Q)
]β
w(Q),
which is (4.4). ¥
Lemma 4.4 (Summation Lemma) Let µ, ν ∈ Z and y ∈ Rn . Then for M > n,
∑
l(Q)=2−µ
(1 +
|y − xQ|2−ν
)−M
≤ cn,M 2(µ−ν)n, if µ ≥ ν. (4.6)
Proof. If µ ≥ ν , i.e., 2−ν ≥ 2−µ , there are 2(µ−ν)n dyadic cubes of size 2−µ in a
dyadic cube of size 2−ν . Fix l ∈ Zn such that y ∈ Qνl . Then the left-hand side of
(4.6) is∑
k∈Zn
(1 + 2ν |y − xQµk|)−M
=∑
i∈Zn
∑
k: Qµk⊆Qν(l+i)
(1 + 2ν |y − xQµk|)−M
≤∑
i∈Zn
(1 + |i|)−M × 2(µ−ν)n ≤ cn 2(µ−ν)n,
again since M > n . ¥
Proof of Theorem 4.2. By definition,
∥∥∥∥∥∑
Q
~sQ mQ
∥∥∥∥∥Bαq
p (W )
=
∥∥∥∥∥∥
∥∥∥∥∥W 1/p∑Q
~sQ (ϕν ∗mQ)
∥∥∥∥∥Lp
ν
∥∥∥∥∥∥lαq
=
∥∥∥∥∥∥
∥∥∥∥∥∥∑
µ∈Z
∑
l(Q)=2−µ
(W 1/p~sQ)(ϕν ∗mQ)
∥∥∥∥∥∥Lp
ν
∥∥∥∥∥∥lαq
.
By Minkowski’s (or the triangle) inequality, the last expression is bounded by
∥∥∥∥∥∥
∑
µ∈Z
∥∥∥∥∥∥∑
l(Q)=2−µ
(W 1/p~sQ)(ϕν ∗mQ)
∥∥∥∥∥∥Lp
ν
∥∥∥∥∥∥lαq
36
≤
∥∥∥∥∥∥∥
∑
µ∈Z
∫
Rn
∑
l(Q)=2−µ
‖W 1/p(x)~sQ‖ |(ϕν ∗mQ)(x)|
p
dx
1/p
ν
∥∥∥∥∥∥∥lαq
=:
∥∥∥∥∥∥
{∑µ>ν
J1/p1 +
∑µ≤ν
J1/p2
}
ν
∥∥∥∥∥∥lαq
. (4.7)
Using estimates (4.2) and (4.3) with θ1 = −(µ− ν)σ , θ2 = −(ν − µ)τ and r1 = 2−ν ,
r2 = 2−µ , we bound each Ji , i = 1, 2:
Ji ≤ c
∫
Rn
∑
l(Q)=2−µ
‖W 1/p(x)~sQ‖H|Q|−1/2 2θi
(1 +
|x− xQ|ri
)−M
p
dx.
If p > 1, split M = M1 +M2 , where M1 > β/p and M2 > n/p′ (this is possible since
M > J ). If p = 1, M = M1 > β (and n/p′ = 0 in further calculations). Then by
the discrete Holder inequality with wQ(x) = ‖W 1/p(x)~sQ‖pH , we get
Ji ≤ cp
∫
Rn
∑
l(Q)=2−µ
wQ(x)|Q|−p/2 2θip
(1 +
|x− xQ|ri
)−M1p
× ∑
l(Q)=2−µ
(1 +
|x− xQ|ri
)−M2p′
p/p′
dx.
By the Summation Lemma 4.4 (with ν = µ in (4.6)), we have
J2 ≤ cp,n2θ2p∑
l(Q)=2−µ
|Q|−p/2
∫
Rn
wQ(x)(1 + 2µ|x− xQ|)−M1p dx,
since M2 > n/p′ . Applying the Squeeze Lemma 4.3 with r = 2−µ = l(Q) and
L = M1p (and so L > β ), we get
J2 ≤ cp,n,β 2−(ν−µ)τp∑
l(Q)=2−µ
|Q|−p/2wQ(Q).
By the Summation Lemma 4.4 (with µ > ν in (4.6)), we have
J1 ≤ cp,n2(ν−µ)(σ−n/p′)p∑
l(Q)=2−µ
|Q|−p/2
∫
Rn
wQ(x)(1 + 2ν |x− xQ|)−M1p dx,
37
again since M2 > n/p′ . Applying the Squeeze Lemma 4.3 again with r = 2−ν >
2−µ = l(Q) and L = M1p , we get
J1 ≤ cp,n,β 2(ν−µ)(σ−n/p′−β/p) p∑
l(Q)=2−µ
|Q|−p/2wQ(Q).
Observe that the last sum is equal to∥∥∥∑
l(Q)=2−µ |Q|−1/2~sQχQ
∥∥∥p
Lp(W ). Combining the
estimates for J1 and J2 (recall that J = np′ + β
p), we have
2να
(∑µ>ν
J1/p1 +
∑µ≤ν
J1/p2
)≤ cp,n,β
∑
µ∈Z2(ν−µ)α
(2(ν−µ)(σ−J)χ{ν−µ<0}
+2−(ν−µ)τχ{ν−µ≥0})× 2µα
∥∥∥∥∥∥∑
l(Q)=2−µ
|Q|−1/2~sQχQ
∥∥∥∥∥∥Lp(W )
. (4.8)
Denote
ai = 2iα(2i(σ−J)χ{i<0} + 2−iτχ{i≥0}
)
and
bµ = 2µα
∥∥∥∥∥∥∑
l(Q)=2−µ
|Q|−1/2 ~sQχQ
∥∥∥∥∥∥Lp(W )
.
Then the right side of (4.8) is nothing else but c∑
µ∈Zaν−µ bµ = c (a∗b)(ν). Substituting
this into (4.7), we get
∥∥∥∥∥∑Q
~sQ mQ
∥∥∥∥∥Bαq
p (W )
≤∥∥∥∥∥∥
{∑
µ∈Z
∑i=1,2
J1/pi
}
ν
∥∥∥∥∥∥lαq
≤ cp,n,β ‖a ∗ b‖lq . (4.9)
Observe that
‖a ∗ b‖lq ≤ ‖a‖l1‖b‖lq for q ≥ 1 (4.10)
and
‖a ∗ b‖lq ≤ ‖a‖lq‖b‖lq for q < 1 (4.11)
(to get the last inequality, apply the q -triangle inequality followed by ‖a ∗ b‖l1 ≤‖a‖l1‖b‖l1 ). For any 0 < q < ∞ , ‖a‖q
lq =∑i<0
2i(σ+α−J)q +∑i≥0
2−i(τ−α)q . Both sums
38
converge, since τ > α and σ + α > J by (4.2) and (4.3). Hence, ‖a‖lq ≤ cq for any
q > 0. (In fact, here we only need 0 < q ≤ 1.) Combining all the estimates together
into (4.9), we obtain
∥∥∥∥∥∑
Q
~sQ mQ
∥∥∥∥∥Bαq
p (W )
≤ c ‖b‖lq = c
∥∥∥∥∥∥∥
2να
∥∥∥∥∥∥∑
l(Q)=2−ν
|Q|−1/2~sQχQ
∥∥∥∥∥∥Lp(W )
ν
∥∥∥∥∥∥∥lq
= c ‖{~sQ}‖bαqp (W ),
where c = cn,p,q,β . ¥
Remark 4.5 Since ψ ∈ A, observe the following properties of ψQ :
1. 0 /∈ supp ψQ for any dyadic Q, and, therefore,∫
xγψQ(x) dx = 0 for any
multi-index γ ;
2. |DγψQ| ≤ cγ,L|Q|− 12− |γ|
n
(1 +
|x− xQ|l(Q)
)−L−|γ|for each L ∈ N ∪ {0} and γ as
before.
Hence, {ψQ}Q is a family of smooth molecules for Bαqp (W ), and for ~f =
∑Q ~sQ ψQ ,
we obtain the boundedness of the inverse ϕ-transform Tψ :
Corollary 4.6 Let W be a doubling matrix of order p, and consider the sequence
~s = {~sQ}Q ∈ bαqp (W ). Then for all 1 ≤ p < ∞, 0 < q ≤ ∞ and α ∈ R,
‖Tψ~s ‖Bαqp (W ) =
∥∥∥∥∥∑Q
~sQ ψQ
∥∥∥∥∥Bαq
p (W )
≤ c ‖{~sQ}Q‖bαqp (W ). (4.12)
In particular, given ~f ∈ Bαqp (W ), consider ~s = Sϕ
~f . Then by (2.2)
∥∥∥~f∥∥∥
Bαqp (W )
=
∥∥∥∥∥∑Q
~sQ ψQ
∥∥∥∥∥Bαq
p (W )
≤ c ‖{~sQ}Q‖bαqp (W ) = c
∥∥∥Sϕ~f∥∥∥
bαqp (W )
.
39
4.2 Decompositions of an exponential type func-
tion
Definition 4.7 For ν ∈ Z, let Eν = {~f : fi ∈ S ′ and supp fi ⊆ {ξ ∈ Rn : |ξ| ≤2ν+1}, i = 1, ..., m}. Then we say that Eν consists of vector functions of exponential
type 2ν+1 .
Consider the following lemma on the decomposition of an exponential type function
(for the proof the reader is referred to [FJW, p.55]):
Lemma 4.8 Suppose g ∈ S ′(Rn), h ∈ S(Rn) and supp g, supp h ⊆ {|ξ| < 2νπ} for
some ν ∈ Z. Then
(g ∗ h)(x) =∑
k∈Zn
2−νng(2−νk) h(x− 2−νk). (4.13)
Note: if h(ξ) = 1 on supp g , then g ∗ h = g .
Let Γ = {γ ∈ S : γ = 1 on {ξ ∈ Rn : |ξ| ≤ 2} and supp γ ⊆ {ξ ∈ Rn : |ξ| < π}} .
Define γν(x) = 2νnγ(2νx) for ν ∈ Z . Since γν = γ(2νξ), supp γν ⊆ {ξ ∈ Rn :
|ξ| < 2νπ} .
Lemma 4.9 For ν ∈ Z let ~g ∈ Eν and fix x ∈ Qνk where k ∈ Zn . Then for any
y ∈ Rn and γ ∈ Γ
~g(y) =∑
l∈Zn
2−νn~g(2−νl + x) γν(y − (2−νl + x)). (4.14)
Proof. Denote ~g x(y) = ~g(y + x). Trivially, ~g(y) = ~g x(y − x). Note that
(~g x)ˆ(ξ) = eixξ~g(ξ), and so supp (~g x)ˆ = supp ~g . Therefore, by (4.13) applied to ~g x :
~g(y) = ~g x(y − x) =∑
l∈Zn
2−νn~g x(2−νl) γν(y − x− 2−νl),
which is (4.14). ¥
40
Lemma 4.10 Let g ∈ S ′ with supp g ⊆ {|ξ| ≤ 3} and γ ∈ Γ. Then for any
x, s ∈ Rn and j ∈ N ∪ {0}
Djg(x + s) =∑
k∈Zn
g(x + k)Djγ(s− k). (4.15)
Proof. Let gx(s) = g(x + s). Observe that supp gx ⊆ {|ξ| ≤ 3} , since gx(ξ) =
eixξg(ξ) and supp g ⊆ {|ξ| ≤ 3} . Applying the decomposition from Lemma (4.8)
(with ν = 0) g(s) =∑
k∈Zn
g(k)γ(s− k) to gx(s), we get
gx(s) =∑
k∈Zn
gx(k)γ(s− k) =∑
k∈Zn
g(x + k)γ(s− k).
Note the following two implications:
1. gx = gx ∗ γ =⇒ Djgx = gx ∗Djγ
2. (Djgx) = gx · (Djγ) =⇒ supp (Djgx) ⊆ supp gx ∩ supp (Djγ) ⊆ {|ξ| ≤ 3}
Therefore,
Djg(x + s) = Djgx(s) =∑
k∈Zn
gx(k)Djγ(s− k) =∑
k∈Zn
g(x + k)Djγ(s− k).
¥
Remark 4.11 Let f ∈ S ′ . Recall the dilations of ϕ: ϕν(x) = 2νnϕ(2νx). Since
supp ϕ ⊆ {ξ ∈ Rn : 12≤ |ξ| ≤ 2}, supp ϕν ⊆ {ξ ∈ Rn : 2ν−1 ≤ |ξ| ≤ 2ν+1} as well
as supp (ϕν ∗ f ) ⊆ {ξ ∈ Rn : 2ν−1 ≤ |ξ| ≤ 2ν+1}. Observe that (ϕν ∗ f) ∈ S ′ and
so (ϕν ∗ f) ∈ Eν . Thus, all previous lemmas apply to ϕν ∗ f .
4.3 Boundedness of the ϕ-transform
Before we talk about the boundedness of the ϕ-transform, we develop two “maximal
operator” type inequalities:
41
Lemma 4.12 Let 1 < p < ∞, W ∈ Ap and ~g ∈ E0 . Then
∑
k∈Zn
∫
Q0k
‖W 1/p(x)~g(k) ‖p dx ≤ cp,n ‖~g ‖pLp(W ). (4.16)
Remark 4.13 Note that in terms of reducing operators, (4.16) is equivalent to
‖{AQ0k~g(k)}k∈Zn‖lp =
(∑
k∈Zn
‖AQ0k~g(k) ‖p
)1/p
≤ cp,n ‖~g ‖Lp(W ). (4.17)
Proof. Let γ ∈ Γ. Then for ~g ∈ E0 , we have ~g = γ ∗ ~g , and the left-hand side
of (4.16) is∑
k∈Zn
∫
Q0k
∥∥∥∥W 1/p(x)
∫
Rn
~g(y)γ(k − y) dy
∥∥∥∥p
dx
≤ cM
∑
k∈Zn
∫
Q0k
(∫
Rn
‖W 1/p(x)~g(y)‖(1 + |k − y|)M
dy
)p
dx,
for some M > n+βp/p′ , where β is the doubling exponent of W , since γ ∈ S . Since
Rn =⋃
m∈Zn
Q0m and mi ≤ yi < mi + 1, i = 1, ..., n , on each Q0m , the last sum is
bounded by
c∑
k∈Zn
∫
Q0k
( ∑
m∈Zn
∫Q0m
‖W 1/p(x)~g(y)‖ dy
(1 + |k −m|)M
)p
dx.
Writing M = M/p + M/p′ and using the discrete Holder inequality (note that M >
n), we bound the last expression by
c∑
k∈Zn
∫
Q0k
∑
m∈Zn
(∫Q0m
‖W 1/p(x)~g(y)‖ dy)p
(1 + |k −m|)Mdx. (4.18)
Observe that
(∫
Q0m
‖W 1/p(x)~g(y)‖ dy
)p
≤(∫
Q0m
‖W 1/p(x)W−1/p(y)‖ ‖W 1/p(y)~g(y)‖ dy
)p
≤(∫
Q0m
‖W 1/p(x)W−1/p(y)‖p′ dy
)p/p′ (∫
Q0m
‖W 1/p(y)~g(y)‖p dy
),
again by Holder’s inequality. By Lemma 3.19, vx(y) = ‖W 1/p(x)W−1/p(y)‖p′ is a
doubling measure with the doubling exponent β :
vx(Q0m) ≤ vx(B(m, |k −m|+√n)) ≤ c (1 + |k −m|)βvx(Q0k).
42
Thus, (4.18) is bounded by
c∑
k,m∈Zn
(1 + |k −m|)βp/p′−M
[∫
Q0k
(∫
Q0k
‖W 1/p(x)W−1/p(y)‖p′dy
)p/p′
dx
]
(4.19)
×∫
Q0m
‖W 1/p(y)~g(y)‖p dy.
By Lemma 1.3, the expression in the square brackets of (4.19) is bounded by a
constant independent of k . Since M > βp/p′ + n , the sum on k converges and,
therefore, (4.19) is estimated above by
c∑
m∈Zn
∫
Q0m
‖W 1/p(y)~g(y)‖p dy = c
∫
Rn
‖W 1/p(y)~g(y)‖p dy = c ‖~g ‖pLp(W ).
¥
Lemma 4.14 Let W be a doubling matrix of order p, 1 ≤ p < ∞, with doubling
exponent β such that p > β , and let ~g ∈ E0 . Then (4.16) holds. Furthermore, if W
is a diagonal matrix, then (4.16) holds for any 1 ≤ p < ∞.
Proof. First, assume (~g)i ∈ S with supp (~g)∧i ⊆ {|ξ| < π} , i = 1, ..., m .
We want to show that for such ~g , the sum on the left-hand side of (4.16) is finite.
Choosing r > β + n , we have
∑
k∈Zn
∫
Q0k
‖W 1/p(x)~g(k)‖p dx ≤∑
k∈Zn
c
(1 + |k|)r
∫
Q0k
‖W 1/p(x)‖p dx.
Since w(x) = ‖W 1/p(x)‖p is a scalar doubling measure, w(Q0k) ≤ c (1 + |k|)βw(Q00).
Hence,
∑
k∈Zn
∫
Q0k
‖W 1/p(x)~g(k)‖p dx ≤∑
k∈Zn
cw(Q00)
(1 + |k|)r−β≤ cw(Q00) < ∞,
since r − β > n .
43
Now we will prove (4.16) for ~g with (~g)i ∈ S and supp (~g)∧i ⊆ {|ξ| ≤ 3} , and then
generalize it to (~g)i ∈ S ′ . Let 0 < δ < 1. Then Bδ(k) ⊆ 3Q0k . Using the doubling
property of wk(x) = ‖W 1/p(x)~g(k)‖p , we “squeeze” each Q0k into Bδ(k):
wk(Q0k) ≤ wk(3Q0k) ≤ c
[ |3Q0k||Bδ(k)|
]β/n
wk(Bδ(k)) ≤ cβ δ−βwk(Bδ(k)).
Hence, the left-hand side of (4.16) is bounded by
cβ δ−β∑
k∈Zn
∫
Bδ(k)
‖W 1/p(x)~g(k)‖p dx. (4.20)
To estimate the integral, we will use the trivial identity ~g(k) = ~g(x) + [~g(k) − ~g(x)]
for x ∈ Bδ(k). Apply the decomposition from Lemma 4.8 with γ ∈ Γ:
~g(k) =∑
m∈Zn
~g(m)γ(k −m) and ~g(x) =∑
m∈Zn
~g(m)γ(x−m).
Using the Mean Value Theorem for [γ(k−m)−γ(x−m)] and the properties of γ ∈ S(note that |x− k| < δ ), we have
‖W 1/p(x)~g(k)‖p ≤ cp‖W 1/p(x)~g(x)‖p + cp,M δp∑
m∈Zn
‖W 1/p(x)~g(m) ‖p
(1 + |k −m|)M,
(4.21)
for some M > β + n . Integrating (4.21) over Bδ(k), we get
∫
Bδ(k)
‖W 1/p(x)~g(k)‖p dx ≤ cp
∫
Bδ(k)
‖W 1/p(x)~g(x)‖p dx
+c δp∑
m∈Zn
∫Bδ(k)
‖W 1/p(x)~g(m) ‖p dx
(1 + |k −m|)M. (4.22)
Apply the doubling property of wm(x) = ‖W 1/p(x)~g(m)‖p again:
wm(Bδ(k)) ≤ wm(B(m, |k −m|+ δ)) ≤ c
[(δ + |k −m|)n
δn
]β/n
wm(Bδ(m))
= c δ−β(1 + |k −m|)βwm(Bδ(m)).
44
Substituting this estimate into (4.22) and summing over k ∈ Zn , we have
∑
k∈Zn
∫
Bδ(k)
‖W 1/p(x)~g(k)‖p dx ≤ cp
∑
k∈Zn
∫
Bδ(k)
‖W 1/p(x)~g(x)‖p dx
+c δp−β∑
m∈Zn
∫
Bδ(m)
‖W 1/p(x)~g(m) ‖p dx
(∑
k∈Zn
(1 + |k −m|)β−M
),
where the last sum converges since M > β + n . If p > β , by choosing 0 < δ < 1/2
such that 1 − c δp−β > 0, we subtract the last term from both sides (note that it is
finite because of our estimates above for ~gi ∈ S ), substitute it into (4.20) and get the
estimate of the left-hand side of (4.16) (note that∑
k∈Zn
∫Bδ(k)
... ≤ ∫Rn ...):
∑
k∈Zn
∫
Q0k
‖W 1/p(x)~g(k)‖p dx ≤ cβ δ−β cp
(1− c δp−β)
∑
k∈Zn
∫
Bδ(k)
‖W 1/p(x)~g(x)‖p dx
≤ cn,β,p
∫
Rn
‖W 1/p(x)~g(x)‖p dx = cn,β,p‖~g ‖pLp(W ). (4.23)
Now let (~g)i ∈ S ′, i = 1, ..., m . Since ~g ∈ E0 , it follows that (~g)i ∈ C∞ , and ~g
and all its derivatives are slowly increasing. Pick a scalar-valued γ ∈ S such that
γ(0) = 1 and supp γ ⊆ B(0, 1). Then for 0 < ε < 1, the function ~g ε(x) := ~g(x)γ(εx)
has its components in S . Observe that (~g ε)∧ = (~g)∧ ∗ [γ(εx)]∧ , with [γ(εx)] ∧ (ξ) =
(1/ε) γ(ξ/ε), and, therefore,
supp (~g ε)∧ ⊆ supp (~g)∧ + supp (1/ε) γ(·/ε) ⊆ {ξ : |ξ| < 3}.
We can apply the result (4.23) to ~g ε :
∑
k∈Zn
∫
Q0k
‖W 1/p(x)~g ε(k)‖p dx ≤ c ‖~g ε‖pLp(W ),
or∑
k∈Zn
∫
Q0k
‖W 1/p(x)~g(k)γ(εk)‖p dx ≤ c
∫
Rn
‖W 1/p(x)~g(x)‖p|γ(εx)|p dx.
Taking lim inf as ε → 0 of both sides and using Fatou’s Lemma on the left-hand side
(with a discrete measure for the sum) and the Dominated Convergence Theorem on
45
the right-hand side, we obtain
∑
k∈Zn
lim infε→0
|γ(εk)|p∫
Q0k
‖W 1/p(x)~g(k)‖p dx
≤ c
∫
Rn
‖W 1/p(x)~g(x)‖p limε→0
|γ(εx)|p dx.
Since γ(εx) −→ε→0
γ(0), we obtain (4.16) for all ~g ∈ E0 .
To get the second assertion of the Lemma, we consider the scalar case with w a
scalar doubling measure. Then (4.22) becomes
w(Bδ(k))|g(k)|p ≤ cp
∫
Bδ(k)
w(x)|g(x)|p dx (4.24)
+cp δpw(Bδ(k))∑
m∈Zn
|g(m)|p(1 + |k −m|)M
,
or
|g(k)|p ≤ cp1
w(Bδ(k))
∫
Bδ(k)
w(x)|g(x)|p dx + c δp∑
m∈Zn
|g(m)|p(1 + |k −m|)M
.
We want to estimate the last sum on m . Fix l ∈ Zn . Dividing everything by
(1 + |k − l|)M and summing on k ∈ Zn , we get
∑
k∈Zn
|g(k)|p(1 + |k − l|)M
≤ cp
∑
k∈Zn
∫Bδ(k)
w(x)|g(x)|p dx
(1 + |k − l|)Mw(Bδ(k))
+c δp∑
k∈Zn
1
(1 + |k − l|)M
∑
m∈Zn
|g(m)|p(1 + |k −m|)M
.
Note that in the last term
∑
k∈Zn
1
(1 + |k − l|)M(1 + |k −m|)M≤ c
(1 + |l −m|)M,
since M > n . Therefore,
∑
k∈Zn
|g(k)|p(1 + |k − l|)M
≤ cp
∑
k∈Zn
∫Bδ(k)
w(x)|g(x)|p dx
(1 + |k − l|)Mw(Bδ(k))+ c δp
∑
m∈Zn
|g(m)|p(1 + |l −m|)M
.
Choose 0 < δ < 1/2 such that 1− c δp > 0. Then
∑
m∈Zn
|g(m)|p(1 + |l −m|)M
≤ cp
1− c δp
∑
m∈Zn
∫Bδ(m)
w(x)|g(x)|p dx
(1 + |l −m|)Mw(Bδ(m)).
46
Substituting this into (4.24) and summing on k ∈ Zn (again using∑
k∈Zn
∫Bδ(k)
... ≤∫Rn ...), we obtain
∑
k∈Zn
w(Bδ(k))|g(k)|p
≤ cp ‖g‖pLp(w) + c δp
∑
k∈Zn
w(Bδ(k))∑
m∈Zn
∫Bδ(m)
w(x)|g(x)|p dx
(1 + |k −m|)Mw(Bδ(m)).
Use the doubling property of w to shift Bδ(k) to Bδ(m). Since δ is fixed, w(Bδ(k)) ≤cδ,n (1 + |k −m|)βw(Bδ(m)), and thus, the last term is dominated by
c δp∑
m∈Zn
∫
Bδ(m)
w(x)|g(x)|p dx×(∑
k∈Zn
(1 + |k −m|)β−M
), (4.25)
where the sum on k converges, since M > β + n . Thus, (4.25) is estimated by
cp,n,β ‖g‖pLp(w) . Hence,
∑
k∈Zn
∫
Q0k
w(x)|g(k)|p dx ≤ cp,n,β
∑
k∈Zn
w(Bδ(k))|g(k)|p ≤ cp,n,β ‖g‖pLp(w).
Now if W is a diagonal matrix, then
‖W 1/p(x) ~u ‖p ≈m∑
i=1
wii(x) |~ui|p,
and thus, applying the scalar case, we get
∑
k∈Zn
∫
Q0k
‖W 1/p(x)~g(k)‖p dx ≈m∑
i=1
∑
k∈Zn
∫
Q0k
wii(x)|~gi(k)|p dx
≤m∑
i=1
c ‖~gi‖pLp(wii)
≈ cp,n,β,m ‖~g‖pLp(W ).
¥
Theorem 4.15 Let α ∈ R, 0 < q ≤ ∞, 1 ≤ p < ∞, and let W satisfy any of
(A1)-(A3). Then
‖{~sQ}Q‖bαqp (W ) ≤ c ‖~f ‖Bαq
p (W ), (4.26)
where ~sQ = Sϕ~f =
⟨~f, ϕQ
⟩for a given ~f .
47
Proof. By definition,
‖{~sQ}Q‖bαqp (W ) =
∥∥∥∥∥∥
∥∥∥∥∥∥∑
l(Q)=2−ν
|Q|−1/2∥∥W 1/p · ~sQ
∥∥H χQ
∥∥∥∥∥∥Lp
ν
∥∥∥∥∥∥lαq
=: ‖{Jν}ν‖lαq. (4.27)
Fix ν ∈ Z . Then Q = Qνk =n∏
i=1
[ki
2ν,ki + 1
2ν
], k ∈ Zn , |Q| = 2−νn , ~sQ =
|Q|1/2(ϕν ∗ ~f )(2−νk) and
Jpν =
∑
l(Q)=2−ν
|Q|−p/2
∫
Q
‖W 1/p(t)~sQ‖p dt
=∑
k∈Zn
∫
Qνk
‖W 1/p(t)(ϕν ∗ ~f )(2−νk)‖p dt.
Let ~fν(x) = ~f(2−νx). Then (ϕν ∗ ~f )(2−νk) = (ϕ ∗ ~fν)(k). We substitute this in the
last integral and note that the change of variables y = 2νt (with Wν(t) := W (2−νt))
will yield
Jpν = 2−νn
∑
k∈Zn
∫
Q0k
‖W 1/pν (t)(ϕ ∗ ~fν)(k)‖p dt. (4.28)
Observe that (ϕ ∗ ~fν)i ∈ S ′ , i = 1, ...,m , and ϕ ∗ ~fν ∈ E0 , since supp ˆϕ ⊆ {ξ ∈Rn : 1
2≤ |ξ| ≤ 2} . Using either Lemma 4.12 or Lemma 4.14 with ~g = ϕ ∗ ~fν and Wν
instead of W (both the Ap condition and the doubling condition are invariant with
respect to dilation), we obtain
Jpν ≤ c 2−νn
∫
Rn
‖W 1/pν (t)(ϕ ∗ ~fν)(t)‖p dt.
Changing variables, we get
Jpν ≤ c
∫
Rn
‖W 1/p(t)(ϕν ∗ ~f )(t)‖p dt = c ‖(ϕν ∗ ~f )‖pLp(W ).
Combining the estimates of Jν for all ν into (4.27), we get
∥∥∥∥{⟨
~f, ϕQ
⟩}Q
∥∥∥∥bαqp (W )
= ‖{~sQ}Q‖bαqp (W ) = ‖{Jν}ν‖lαq
48
≤ c
∥∥∥∥{∥∥∥
(ϕν ∗ ~f
)∥∥∥Lp(W )
}
ν
∥∥∥∥lαq
= c∥∥∥~f
∥∥∥Bαq
p (W,∼ϕ)
, (4.29)
where c = c(p, β, n).
To finish the proof of the theorem, we have to establish the equivalence between
Bαqp (W,ϕ) and Bαq
p (W, ϕ). As we mentioned in Section 2, ϕ ∈ A , and so the pair
(ϕ, ψ) satisfies (2.1), since ˆϕ = ¯ϕ and ˆψ =¯ψ . By (2.2), ~f =
∑Q
⟨~f, ϕQ
⟩ψQ . Since
{ψQ}Q is a family of smooth molecules for Bαqp (W ) (see Remark 4.5), by Theorem
4.2 we have
‖~f ‖Bαq
p (W,∼ϕ)≤ c
∥∥∥∥{⟨
~f, ϕQ
⟩}Q
∥∥∥∥bαqp (W )
. (4.30)
Applying (4.29) to the right-hand side of the last inequality, we bound it by
c∥∥∥~f
∥∥∥Bαq
p (W,
∼∼ϕ )
= c∥∥∥~f
∥∥∥Bαq
p (W, ϕ). (4.31)
Finally, combining (4.29) with (4.30) and (4.31), we obtain
∥∥∥∥{⟨
~f, ϕQ
⟩}Q
∥∥∥∥bαqp (W )
≡ ‖{~sQ}Q‖bαqp (W ) ≤ c
∥∥∥~f∥∥∥
Bαqp (W, ϕ)
.
¥
Remark 4.16 The fact that ϕ and ϕ were interchanged in the last step of the previ-
ous theorem can be generalized into Theorem 1.8 about the independence of the space
Bαqp (W ) from the choice of ϕ:
Proof of Theorem 1.8. Let {ϕ(1), ψ(1)} and {ϕ(2), ψ(2)} be two different sets
of mutually admissible kernels. Decompose ~f in the second system:
~f =∑
Q
⟨~f, ϕ
(2)Q
⟩ψ
(2)Q =
∑Q
~s(2)Q ψ
(2)Q .
Observe that ψ(2)Q is a molecule for Q and, therefore, by Theorem 4.2,
‖~f ‖Bαqp (W,ϕ(1)) ≤ c ‖{~s (2)
Q }Q‖bαqp (W ) ≤ c ‖~f ‖Bαq
p (W,ϕ(2)),
49
where the last inequality holds by Theorem 4.15. Interchanging ϕ(1) with ϕ(2) , we get
the norm equivalence between Bαqp (W,ϕ(1)) and Bαq
p (W,ϕ(2)). In other words, the
space Bαqp (W ) is independent of the choice of ϕ under any of the three assumptions
on W . ¥
Remark 4.17 Combining boundedness of the ϕ-transform (Theorem 4.15) and that
of the inverse ϕ-transform (Corollary 4.6), we get the norm equivalence claimed in
Theorems 1.4 and 1.6.
4.4 Connection with reducing operators
Now we connect the weighted sequence Besov space with its reducing operator equiv-
alent. Recall that for each matrix weight W , we can find a sequence of reducing
operators {AQ}Q such that for all ~u ∈ H ,
ρp,Q(~u) =
(1
|Q|∫ ∥∥W 1/p(t) · ~u
∥∥p
H χQ(t) dt
)1/p
≈ ‖AQ~u ‖H. (4.32)
Lemma 4.18 Let α ∈ R, 0 < q ≤ ∞, 1 ≤ p < ∞, and let {AQ}Q be reducing
operators for W . Then
‖{~sQ}Q‖bαqp (W ) ≈ ‖{~sQ}Q‖bαq
p ({AQ}). (4.33)
Proof. Using (4.32), we get the equivalence
‖{~sQ}Q‖bαqp (W ) =
∥∥∥∥∥∥
∥∥∥∥∥∥∑
l(Q)=2−ν
|Q|− 12‖W 1/p · ~sQ‖HχQ
∥∥∥∥∥∥Lp
ν
∥∥∥∥∥∥lαq
=
∥∥∥∥∥∥∥
∑
l(Q)=2−ν
|Q|− p2 [ρp,Q(~sQ)]p |Q|
1p
ν
∥∥∥∥∥∥∥lαq
50
≈
∥∥∥∥∥∥∥
∑
l(Q)=2−ν
|Q|− p2‖AQ~sQ‖p
H
∫χQ(t) dt
1p
ν
∥∥∥∥∥∥∥lαq
=
∥∥∥∥∥∥
∥∥∥∥∥∥∑
l(Q)=2−ν
|Q|− 12‖AQ~sQ‖HχQ
∥∥∥∥∥∥Lp
ν
∥∥∥∥∥∥lαq
= ‖{~sQ}Q‖bαqp ({AQ}).
¥
Finally, combining Theorems 1.4 and 1.6 with (4.33), we get Theorem 1.9.
Corollary 4.19 The space Bαqp (W ) is complete when α ∈ R, 0 < q ≤ ∞, 1 ≤ p <
∞ and W satisfies any of (A1)-(A3).
Proof. If{
~fn
}n∈N
is Cauchy in Bαqp (W ), then
{{~sQ
(~fn
)}Q
}
n∈Nis Cauchy
in bαqp ({AQ}) by Theorem 4.15 and Lemma 4.18 (or just Theorem 1.9). This implies
that ∥∥∥∥∥∥∑
l(Q)=2−ν
|Q|− 12
∥∥∥AQ
[~sQ
(~fn
)− ~sQ
(~fm
)]∥∥∥H
χQ
∥∥∥∥∥∥
p
Lp
= 2νn(p/2−1)∑
l(Q)=2−ν
∥∥∥AQ
[~sQ
(~fn
)− ~sQ
(~fm
)]∥∥∥p
H−→
n,m→∞0,
for each ν ∈ Z . Hence,∥∥∥AQ
[~sQ
(~fn
)− ~sQ
(~fm
)] ∥∥∥H
−→n,m→∞
0 for each Q . Since
the AQ ’s are invertible,{~sQ
(~fn
)}n∈N
is a vector-valued Cauchy sequence in H for
each Q . Therefore, we can define ~sQ = limn→∞
~sQ(~fn). Set ~f =∑
Q ~sQ ψQ . Observe
that∥∥∥~fn − ~f
∥∥∥Bαq
p (W )=
∥∥∥∥∥∑Q
[~sQ
(~fn
)− ~sQ
]ψQ
∥∥∥∥∥Bαq
p (W )
≤ c
∥∥∥∥{~sQ
(~fn
)− ~sQ
}Q
∥∥∥∥bαqp ({AQ})
≤ c lim infm→∞
∥∥∥∥{~sQ
(~fn
)− ~sQ
(~fm
)}Q
∥∥∥∥bαqp ({AQ})
−→n→∞
0,
51
by Corollary 4.6 and Lemma 4.18, the discrete version of Fatou’s Lemma and the fact
that
{ {~sQ
(~fn
)}Q
}
n∈Nis Cauchy in bαq
p ({AQ}). Furthermore, ~f = (~f − ~fn) + ~fn ∈Bαq
p (W ). Thus, Bαqp (W ) is complete. ¥
Recall (Chapter 3) the Ap condition in terms of reducing operators: ‖AQA#Q‖ ≤ c
for any cube Q ∈ Rn ; in other words, ‖AQ y‖ ≤ c ‖(A#Q)−1 y‖ holds for any y ∈
H . Also, the inverse inequality ‖(AQ A#Q)−1‖ ≤ c (or, equivalently, ‖(A#
Q)−1 y‖ ≤c ‖AQ y‖ for any y ∈ H) holds automatically. This implies the following imbeddings
of the sequence Besov spaces:
Corollary 4.20 For α ∈ R, 1 < p < ∞, 0 < q ≤ ∞, and W a matrix weight with
corresponding reducing operators AQ and A#Q ,
1. bαqp ({AQ}) ⊆ bαq
p ({(A#Q)−1}) always,
2. bαqp ({(A#
Q)−1}) ⊆ bαqp ({AQ}) if W ∈ Ap .
52
CHAPTER 5
Calderon-Zygmund Operators on
Matrix-Weighted Besov Spaces
5.1 Almost diagonal operators
Consider bαqp (W ) with parameters α, p, q fixed (α ∈ R , 1 ≤ p < ∞ , 0 < q ≤ ∞) and
W a doubling matrix of order p with doubling exponent β . Also, if p = 1, then the
convention is that 1/p′ = 0.
Definition 5.1 A matrix A = (aQP )Q,P∈D is almost diagonal, A ∈ adαqp (β), if there
exist M > J = np′ + β
pand c > 0 such that for all Q,P ,
|aQP | ≤ c min
([l(Q)
l(P )
]α1
,
[l(P )
l(Q)
]α2)(
1 +|xQ − xP |
max(l(Q), l(P ))
)−M
, (5.1)
with α1 > α + n2
and α2 > J − (α + n2).
Remark 5.2 This definition differs from the definition of almost diagonality in
[FJW], since both α2 and M depend on the doubling exponent β .
To simplify notation for the matrix A above, we will only write (aQP ) without
specifying indices Q,P .
53
Example 5.3 (An almost diagonal matrix) Let ϕ ∈ A. If {mQ}Q is a family
of smooth molecules for Bαqp (W ), then
(aQP ) ∈ adαqp (β), (5.2)
where aQP = 〈mP , ϕQ〉, by (4.2) and (4.3), since 〈mP , ϕQ〉 = |Q|1/2(ϕν ∗mP )(xQ) if
l(Q) = 2−ν .
Now we show that almost diagonal matrices are bounded on bαqp (W ), i.e., Theorem
1.10. First we need the following approximation lemma:
Lemma 5.4 Let P,Q be dyadic cubes and t ∈ Q. Then
1 +|xQ − xP |
max(l(Q), l(P ))≈(n)
1 +|t− xP |
max(l(Q), l(P )). (5.3)
Proof. First suppose that l(Q) ≥ l(P ). If P ⊆ 3Q , then 0 ≤ |xP − xQ| ≤2√
n l(Q) = c l(Q) and so
1 ≤ 1 +|xQ − xP |
l(Q)≤ 1 + c ⇐⇒ 1 +
|xQ − xP |l(Q)
≈ 1.
Also 0 ≤ |xP − t| ≤ 2√
n l(Q) = c l(Q) and thus
1 ≤ 1 +|t− xP |
l(Q)≤ 1 + c ⇐⇒ 1 +
|t− xP |l(Q)
≈ 1,
and (5.3) follows.
If P ∩ 3Q = ∅ , then |xP − t| ≥ l(Q) and |xP − xQ| ≥ l(Q). Since |xQ − t| ≤ c l(Q),
by the triangle inequality we get both
|xP −xQ| ≤ |xP −t|+ |t−xQ| ≤ |xP −t|+c l(Q) ≤ |xP −t|+c |xP −t| ≤ (1+c) |xP −t|
and
|xP − t| ≤ |xP − xQ|+ |xQ − t| ≤ |xP − xQ|+ c l(Q) ≤ (1 + c) |xP − xQ|.
54
Therefore, |xP − xQ| ≈ |xP − t| .Now assume that l(Q) < l(P ). Choose P dyadic with l(P ) = l(P ) and Q ⊆ P .
If P ∩ 3P = ∅ , then |xP − t| ≥ c l(Q) and |xP − xQ| ≥ c l(Q). Hence,
|xP −xQ| ≤ |xP −t|+ |t−xQ| ≤ |xP −t|+c l(Q) ≤ |xP −t|+c |xP −t| ≤ (1+c)|xP −t|,
and
|xP − t| ≤ |xP − xQ|+ |xQ − t| ≤ |xP − xQ|+ c l(Q) ≤ (1 + c)|xP − xQ|,
and we again get |xP − xQ| ≈ |xP − t| .If P ⊆ 3P , then 0 ≤ |xQ − xP | ≤ c1 l(P ) = c1 l(P ) and 0 ≤ |t − xP | ≤ c2 l(P ) =
c2 l(P ); thus,
1 ≤ 1 +|xQ − xP |
l(P )≤ 1 + c1 and 1 ≤ 1 +
|t− xP |l(P )
≤ 1 + c2
which means
1 +|xQ − xP |
l(P )≈ 1 ≈ 1 +
|t− xP |l(P )
.
¥
Proof of Theorem 1.10. Let A = (aQP ) with A ∈ adαqp (β). We want to show
that∥∥∥∥∥∥
{∑P
aQP~sP
}
Q
∥∥∥∥∥∥bαqp (W )
≤ cn,p,q,β ‖{~sQ}Q‖bαqp (W ). (5.4)
By definition, ∥∥∥∥∥∥
{∑P
aQP~sP
}
Q
∥∥∥∥∥∥bαqp (W )
≤
∥∥∥∥∥∥∥
∑
l(Q)=2−ν
|Q|−p/2
∫
Q
(∑P
|aQP |‖W 1/p(t)~sP‖)p
dt
1/p
ν
∥∥∥∥∥∥∥lαq
55
=:
∥∥∥∥∥∥∥
2να 2νn/2
∑
l(Q)=2−ν
JQ
1/p
ν
∥∥∥∥∥∥∥lq
. (5.5)
Substituting the estimate (5.1) for aQP in JQ , we get
JQ ≤ cp,M
∫
Q
∑
j≥0
2−jα2
∑
l(P )=2−(ν+j)
‖W 1/p(t)~sP‖ (1 + 2ν |xQ − xP |)−M
p
dt
+cp,M
∫
Q
∑
j<0
2jα1
∑
l(P )=2−(ν+j)
‖W 1/p(t)~sP‖(1 + 2(ν+j)|xQ − xP |
)−M
p
dt.
Pick ε > 0 sufficiently small such that (i) α1− ε > α+n/2, (ii) α2− ε > J −α−n/2
and (iii) M > β/p + (n + ε)/p′ . Apply the discrete Holder inequality twice, first
with αi = ε + (αi − ε) for the sum on j (note that α1, α2 > 0) and second with
M = n+εp′ +
(M − n+ε
p′
)for the sum on P (if p′ = ∞ , then the Lp′ -norm is replaced
by the supremum):
JQ ≤ cp,M
∫
Q
(∑j≥0
2−jεp′)p/p′ [∑
j≥0
2−j(α2−ε)p
× ∑
l(P )=2−(ν+j)
‖W 1/p(t)~sP‖ (1 + 2ν |xQ − xP |)−M
p dt
+cp,M
∫
Q
(∑j<0
2jεp′)p/p′ [∑
j<0
2j(α1−ε)p
× ∑
l(P )=2−(ν+j)
‖W 1/p(t)~sP‖(1 + 2(ν+j)|xQ − xP |
)−M
p dt
≤ cp,M,ε
∑j≥0
2−j(α2−ε)p
∑
l(P )=2−(ν+j)
(1 + 2ν |xQ − xP |)−n−ε
p/p′
×∑
l(P )=2−(ν+j)
∫
Q
‖W 1/p(t)~sP‖p (1 + 2ν |xQ − xP |)−(M−n+εp′ )p
dt
+cp,M,ε
∑j<0
2j(α1−ε)p
∑
l(P )=2−(ν+j)
(1 + 2(ν+j)|xQ − xP |
)−n−ε
p/p′
56
×∑
l(P )=2−(ν+j)
∫
Q
‖W 1/p(t)~sP‖p(1 + 2(ν+j)|xQ − xP |
)−(M−n+εp′ )p
dt.
Use the Summation Lemma 4.4 to estimate the square brackets and denote wP (t) =
‖W 1/p(t)~sP‖p . By Lemma 5.4, xQ can be replaced by any t ∈ Q , and so we get
JQ ≤ cp,M
∑j≥0
2−j(α2−ε)p+jnp/p′
×∑
l(P )=2−(ν+j)
∫
Q
wP (t)
(1 +
|t− xP |l(Q)
)−(M−n+εp′ )p
dt
+cp,M
∑j<0
2j(α1−ε)p∑
l(P )=2−(ν+j)
∫
Q
wP (t)
(1 +
|t− xP |l(P )
)−(M−n+εp′ )p
dt.
Summing on Q and applying the Squeeze Lemma 4.3 (recall M > β/p + (n + ε)/p′ ),
we get∑
l(Q)=2−ν
JQ ≤ cp,n
∑j≥0
2−j(α2−ε)p+jnp/p′
×∑
l(P )=2−(ν+j)
∑
l(Q)=2−ν
∫
Q
wP (t) (1 + 2ν |t− xP |)−(M−n+εp′ )p
dt
+cp,n
∑j<0
2j(α1−ε)p∑
l(P )=2−(ν+j)
∑
l(Q)=2−ν
∫
Q
wP (t)(1 + 2ν+j|t− xP |
)−(M−n+εp′ )p
dt
≤ cp,n,β
∑
j∈Z
(2−j(α2−ε)p+jnp/p′+jβχ{j≥0} + 2j(α1−ε)pχ{j<0}
) ∑
l(P )=2−(ν+j)
wP (P ).
Observe that 2νnp/2 = |P |−p/22−jnp/2 for l(P ) = 2−(ν+j) , and
∑
l(P )=2−(ν+j)
|P |−p/2wP (P ) =
∥∥∥∥∥∥∑
l(P )=2−(ν+j)
|P |−1/2~sP χP
∥∥∥∥∥∥
p
Lp(W )
.
Then, using 1 < p < ∞ to take the power 1/p inside the sum on j , we get
2να 2νn/2
∑
l(Q)=2−ν
JQ
1/p
≤ c∑
j∈Z
[2−jα2−jn/2
(2−j(α2−ε)p+jnp/p′+jβχ{j≥0}
+2j(α1−ε)pχ{j<0})1/p
]× 2(ν+j)α
∥∥∥∥∥∥∑
l(P )=2−(ν+j)
|P |−1/2~sP χP
∥∥∥∥∥∥Lp(W )
57
=: c∑
j∈Za−j × bν+j = c (a ∗ b)(ν).
Use (4.10) and (4.11) to estimate the norm of the convolution ‖a ∗ b‖lq . Then for
q ≤ 1,
‖a‖qlq =
∑j≤0
2j(α+n/2+(α2−ε)−J)q +∑j>0
2j(α+n/2−(α1−ε))q ≤ cq,
since α1 − ε > α + n/2 and α2 − ε > J − (α + n/2). Using the ‖a‖l1 estimate for
q ≥ 1 and the ‖a‖lq estimate for q < 1, and substituting into (5.5), we obtain
∥∥∥∥∥∥
{∑P
aQP~sP
}
Q
∥∥∥∥∥∥bαqp (W )
≤ c ‖b‖lq = c
∥∥∥∥∥∥∥
2µα
∥∥∥∥∥∥∑
l(P )=2−µ
|P |−1/2~sP χP
∥∥∥∥∥∥Lp(W )
µ
∥∥∥∥∥∥∥lq
= c ‖{~sP}P‖bαqp (W ),
where c = cn,p,q,β . ¥
Now we will show that the class of almost diagonal matrices is closed under com-
position. For ε > 0, δ > 0, J = np′ + β
pand P, Q ∈ D , denote
wQP (δ, ε) =
[l(Q)
l(P )
]α+n2
min
([l(Q)
l(P )
] ε2
,
[l(P )
l(Q)
] ε2+J
)(1 +
|xQ − xP |max(l(Q), l(P ))
)−J−δ
.
Theorem 5.5 Let A,B ∈ adαqp (β). Then A ◦B ∈ adαq
p (β).
We need the following lemma, which is a modification of [FJ2, Theorem D.2]
adjusted to the weighted ad condition:
Lemma 5.6 Let δ, γ1, γ2 > 0, γ1 6= γ2 , and 2δ < γ1 + γ2 . Then there exists a
constant c = cn,δ,γ1,γ2,J such that
∑R
wQR(δ, γ1) wRP (δ, γ2) ≤ cwQP (δ, min(γ1, γ2)). (5.6)
Proof. Without loss of generality, we may assume that α = −n/2, since the
terms [l(R)]α+n/2 cancel in the sum of (5.6), leaving[
l(Q)l(P )
]α+n/2
for the right-hand side
58
of the inequality. Denote γ = min(γ1, γ2). With l(Q) = 2−q, l(P ) = 2−p, l(R) = 2−r ,
first assume l(P ) ≤ l(Q). Then the sum in (5.6) can be split into the following terms:
∑
l(R)<l(P )≤l(Q)
+∑
l(P )≤l(R)≤l(Q)
+∑
l(P )≤l(Q)<l(R)
= I + II + III.
Then
I =∑
R
[l(R)
l(Q)
]γ1/2+J (1 +
|xQ − xR|l(Q)
)−J−δ [l(R)
l(P )
]γ2/2 (1 +
|xR − xP |l(Q)
)−J−δ
= [l(Q)]−(γ1/2+J)[l(P )]−γ2/2
∞∑r=p+1
2−r(γ1+γ2
2+J)gP,Q,J+δ,r(xP )
≤ c [l(Q)]−(γ1/2+J)[l(P )]−γ2/2+n
(1 +
|xQ − xP |l(Q)
)−J−δ ∞∑r=p+1
2−r(γ1+γ2
2+J−n),
by [FJ2, Lemma D.1]. Since J > n , the geometric progression sum is bounded by
c 2−p(γ1+γ2
2+J−n) = c [l(P )](
γ1+γ22
+J−n) . Thus,
I ≤ c
[l(P )
l(Q)
]γ1/2+J (1 +
|xQ − xP |l(Q)
)−J−δ
≤ c
[l(P )
l(Q)
]γ/2+J (1 +
|xQ − xP |l(Q)
)−J−δ
,
substituting γ1 with γ , since l(P ) ≤ l(Q).
Similarly, using [FJ2, Lemma D.1], we have
II =∑
R
[l(R)
l(Q)
]γ1/2+J (1 +
|xQ − xR|l(Q)
)−J−δ [l(P )
l(R)
]γ2/2+J (1 +
|xR − xP |l(R)
)−J−δ
≤ [l(P )]γ2/2+J
[l(Q)]γ1/2+J
p∑r=q
2−r(γ1−γ2
2)gP,Q,J+δ,r(xP )
≤ c
(1 +
|xQ − xP |l(Q)
)−J−δ[l(P )]γ2/2+J
[l(Q)]γ1/2+J[l(P )]
γ1−γ22
= c
[l(P )
l(Q)
]γ1/2+J (1 +
|xQ − xP |l(Q)
)−J−δ
,
and γ1 can be replaced by γ , since l(P ) ≤ l(Q).
The estimate of III is also similar:
III ≤ [l(Q)]γ1/2[l(P )]γ2/2+J
q−1∑r=−∞
2r(γ1+γ2
2+J)gP,Q,J+δ,r(xP )
59
≤ [l(Q)]γ1/2[l(P )]γ2/2+J
q−1∑r=−∞
2r(γ1+γ2
2+J)
(1 +
|xQ − xP |l(R)
)−J−δ
.
Observe that
(1 +
|xQ − xP |l(R)
)J+δ
>
(l(Q)
l(R)+
l(Q)
l(R)
|xQ − xP |l(Q)
)J+δ
>
[l(Q)
l(R)
]J+δ (1 +
|xQ − xP |l(Q)
)J+δ
.
Then
III ≤ c [l(Q)]γ1/2−J−δ[l(P )]γ2/2+J
(1 +
|xQ − xP |l(Q)
)−J−δ q−1∑r=−∞
2r(γ1+γ2
2+J)2r(−J−δ),
where the last sum converges since γ1+γ2
2> δ and is bounded by c 2q(
γ1+γ22
−δ) . Sim-
plifying, we get
III ≤ c
[l(P )
l(Q)
]γ2/2+J (1 +
|xQ − xP |l(Q)
)−J−δ
.
Combining I, II and III , we get the right-hand side estimate of (5.6), if l(P ) ≤ l(Q).
The case l(P ) > l(Q) follows by exact repetition of the steps above. ¥
Proof of Theorem 5.5. Since A = (aQP ), B = (bQP ) ∈ adαqp (β), for each
i = A, B there exist 0 < εi < min(α1−(α+n/2), α2−J+α+n/2) and 0 < δ < M−J
such that |aQP | ≤ cwQP (δ, εA) and |bQP | ≤ cwQP (δ, εB). Without loss of generality,
we may assume εA < εB and δ <εA + εB
2. Then
|(AB)QP | ≤ |∑
R
aQR bRP | ≤ c∑
R
wQR(δ, εA) wRP (δ, εB) ≤ cwQP (δ, εA),
by Lemma 5.6, which means that A ◦B ∈ adαqp (β). ¥
Definition 5.7 Let T be a continuous linear operator from S to S ′ . We say that
T is an almost diagonal operator for Bαqp (W ), and write T ∈ ADαq
p (β), if for some
pair of mutually admissible kernels (ϕ, ψ), the matrix (aQP ) ∈ adαqp (β), where aQP =
〈TψP , ϕQ〉.
60
Remark 5.8 The definition of T ∈ ADαqp (β) is independent of the choice of the pair
(ϕ, ψ).
Proof. Define S0 = {f ∈ S : 0 /∈ supp f} . Observe that ψ ∈ A implies
ψ, ψν , ψQ ∈ S0 for ν ∈ Z and Q dyadic. Moreover, if g ∈ S0 , then gN :=N∑
ν=−N
ϕν ∗ψν ∗ g converges to g as N → ∞ in the S -topology (for proof refer to Appendix,
Lemma A.1). Since T is continuous from S into S ′ , we have Tg =∑
ν∈ZT (ϕν ∗
ψν ∗ g). Furthermore, for g ∈ S0 and fixed ν ∈ Z , we have∑
|k|≤M
〈g, ϕQνk〉ψQνk
converges to ϕν ∗ ψν ∗ g as M → ∞ in the S -topology (again refer to Appendix,
Lemma A.2). Hence, Tg =∑
ν∈Z
∑
k∈Zn
〈g, ϕQνk〉 TψQνk
=∑Q
〈g, ϕQ〉 TψQ . Now,
suppose (〈TψP , ϕQ〉QP ) ∈ adαqp (β) for some fixed pair (ϕ, ψ) of mutually admissible
kernels. Take any other such pair (ϕ, ψ). Then ψP =∑
L
⟨ψP , ϕL
⟩ψL and ϕQ =
∑R
〈ϕQ, ψR〉ϕR , which gives
⟨T ψP , ϕQ
⟩=
∑R,L
⟨ψP , ϕL
⟩〈TψL, ϕR〉 〈ϕQ, ψR〉.
Since both {ψR}R and {ϕL}L constitute families of smooth molecules for Bαqp (W ),
by (5.2) the matrices(⟨
ψP , ϕL
⟩LP
), (〈ϕQ, ψR〉QR) ∈ adαq
p (β). By Theorem 5.5,(⟨T ψP , ϕQ
⟩QP
)∈ adαq
p (β). ¥
A straightforward consequence of Theorem 1.10 is the following statement:
Corollary 5.9 Let T ∈ ADαqp (β), α ∈ R, 1 ≤ p < ∞, 0 < q < ∞. Then T
extends to a bounded operator on Bαqp (W ) if W satisfies any of (A1)-(A3).
Proof. First, consider ~f with(
~f)
i∈ S0 . Let (ϕ, ψ) be a pair of mutually admis-
sible kernels. Denote ~tQ =∑
P 〈TψP , ϕQ〉~sP (~f ) and observe that (〈TψP , ϕQ〉QP ) ∈adαq
p (β). Using the ϕ-transform decomposition ~f =∑
P ~sP (~f ) ψP and taking T
61
inside the sum as in the previous remark, we get
‖T ~f ‖Bαqp (W ) =
∥∥∥∥∥∑
P
~sP (~f )TψP
∥∥∥∥∥Bαq
p (W )
=
∥∥∥∥∥∑Q
(∑P
〈TψP , ϕQ〉~sP (~f )
)ψQ
∥∥∥∥∥Bαq
p (W )
= ‖∑Q
~tQψQ‖Bαqp (W ) ≤ c ‖{~tQ}Q‖bαq
p (W )
≤ c ‖{~sQ}Q‖bαqp (W ) ≤ c ‖~f ‖Bαq
p (W ),
by Corollary 4.6, Theorem 1.10 and Theorem 4.15.
Note that S0 is dense in Bαqp (W ) if α ∈ R , 0 < q < ∞ , 1 ≤ p < ∞ and W
satisfies any of (A1)-(A3) (for the proof, refer to Appendix). Thus, T extends to all
of Bαqp (W ). ¥
Note that if q = ∞ , then T extends to a bounded operator on the closure of S0
in Bα∞p (W ).
Remark 5.10 Let {mQ}Q be a family of smooth molecules for Bαqp (W ). Apply the
ϕ-transform to∑
P
~sP mP :
~tQ := Sϕ
(∑P
~sP mP
)=
⟨∑P
~sP mP , ϕQ
⟩=
∑P
〈mP , ϕQ〉 ~sP .
Then(〈mP , ϕQ〉QP
)forms an almost diagonal matrix by (5.2), and therefore, by
Theorem 1.10,
‖{~tQ}Q‖bαqp (W ) ≡ ‖Sϕ(
∑P
~sP mP )‖bαqp (W ) ≤ c ‖{~sP}P‖bαq
p (W ),
if W is doubling.
Corollary 5.11 Let T, S ∈ ADαqp (β). Then T ◦ S ∈ ADαq
p (β).
62
Proof. Since T, S ∈ ADαqp (β), it follows that (tQP ) := (〈TψP , ϕQ〉QP ) is in
adαqp (β), and so is (sQP ) := (〈SψP , ϕQ〉QP ). Thus, for Q,P dyadic we have SψP =
∑R
〈SψP , ϕR〉ψR , and so
〈T ◦ SψP , ϕQ〉 =∑
R
〈SψP , ϕR〉 〈TψR, ϕQ〉 =∑
R
tQR sRP ∈ adαqp (β),
by Theorem 5.5 (composition of almost diagonal matrices). ¥
5.2 Calderon-Zygmund operators
In this section we show that Calderon-Zygmund operators (CZOs) are bounded on
Bαqp (W ) for certain parameters α, p, q, β . First we recall the definition of smooth
atoms and the fact that a CZO maps smooth atoms into smooth molecules. Then we
use a general criterion for boundedness of operators: if an operator T maps smooth
atoms into molecules, then its matrix (〈TψP , ϕQ〉QP ) forms an almost diagonal oper-
ator on bαqp (W ), and therefore, T is bounded on Bαq
p (W ).
Definition 5.12 Let N ∈ N ∪ {0}. A function aQ ∈ D(Rn) is a smooth N -atom
for Q if
1. supp aQ ⊆ 3Q,
2.
∫xγaQ(x) dx = 0 for |γ| ≤ N , and
3. |DγaQ(x)| ≤ cγ l(Q)−|γ|−n/2 for all |γ| ≥ 0.
Let 0 < δ ≤ 1, M > 0, N ∈ N ∪ {0,−1} , N0 ∈ N ∪ {0} .
Lemma 5.13 (Boundedness criterion) Suppose a continuous linear operator
T : S → S ′ maps any smooth N0 -atom into a fixed multiple of a smooth (δ,M,N)-
molecule for Bαqp (W ), α ∈ R, 1 ≤ p < ∞, 0 < q ≤ ∞ with δ,M,N satisfying (M.i),
63
(M.ii) and (M.iii) (see Section 4.1). Suppose W satisfies any of (A1)-(A3). Then
T ∈ ADαqp (β) and, if q < ∞, T extends to a bounded operator on Bαq
p (W ).
Proof. By Corollary 5.9, it suffices to show that (〈TψP , ϕQ〉QP ) ∈ adαqp (β) for
some ϕ, ψ ∈ A satisfying (2.1). Observe that if ψ ∈ A , then there exists θ ∈ S with
supp θ ⊆ B1(0),∫
xγθ(x) dx = 0, if |γ| ≤ N0 , and∑
ν∈Zθ(2−νξ)ϕ(2−νξ) = 1 for ξ 6= 0
([FJW, Lemma 5.12]). Using ψP =∑
ν∈Zθν ∗ ϕν ∗ ψP as in the atomic decomposition
theorem ([FJW, Theorem 5.11]), we have
ψP (x) =∑Q
tQP a(P )Q (x) (5.7)
with tQP = |Q|1/2 supy∈Q
|(ϕν ∗ ψP )(y)| for l(Q) = 2−ν , and each a(P )Q is an N0 -atom
defined by
a(P )Q (x) =
1
tQP
∫
Q
θν(x− y) (ϕν ∗ ψP )(y) dy if tQP 6= 0 (5.8)
and a(P )Q = 0 if tQP = 0. Using (4.2)-(4.3) (valid because {ψP}P is a family of
molecules for Bαqp (W )), we get
|(ϕν ∗ ψP )(y)| ≤ c |P |−1/2 min
([l(Q)
l(P )
]τ
,
[l(P )
l(Q)
]σ)(1 +
|y − xP |max(l(Q), l(P ))
)−M
,
for some τ > α and σ > J − α . In fact, ϕν ∗ ψP = 0 if |µ − ν| > 1 (2−µ = l(P )),
since ϕ, ψ ∈ A , but all we require is the previous estimate. Since y ∈ Q , y can be
replaced by xQ in the last expression by Lemma 5.4, and so
|tQP | ≤ c
( |Q||P |
)1/2
min
([l(Q)
l(P )
]τ
,
[l(P )
l(Q)
]σ)(1 +
|xQ − xP |max(l(Q), l(P ))
)−M
,
which is exactly (5.1). Thus (tQP ) ∈ adαqp (β). Using (5.7), we obtain
〈TψP , ϕQ〉 =
⟨∑R
tRP Ta(P )R , ϕQ
⟩=
∑R
tRP
⟨Ta
(P )R , ϕQ
⟩.
64
Since T maps any N0 -atom a(P )R into a fixed multiple of a smooth (δ,M, N)-molecule
mR : Ta(P )R = cmR and c depends neither on R nor on Q , we get
⟨Ta
(P )R , ϕQ
⟩= c 〈mR, ϕQ〉 =: c tQR,
and by (5.2), since mR is a smooth (δ,M,N)-molecule for Bαqp (W ), (tQR) ∈ adαq
p (β).
Hence,(〈TψP , ϕQ〉QP
)=
(c
∑R
tQR tRP
)∈ adαq
p (β),
since the composition of two almost diagonal operators is again almost diagonal by
Theorem 5.11. ¥
Let T be a continuous linear operator from S(Rn) to S ′(Rn), and let K = K(x, y)
be its distributional kernel defined on R2n�∆, where ∆ = {(x, y) ∈ Rn×Rn : x = y}(for definitions refer to [FJW], Chapter 8). Then T ∈ CZO(ε), 0 < ε ≤ 1, if K has
the following properties:
(I) |K(x, y)| ≤ c
|x− y|n ,
(IIε ) |K(x, y)−K(x′, y)|+ |K(y, x)−K(y, x′)| ≤ c|x− x′|ε|x− y|n+ε
if 2|x− x′| ≤ |x− y| .
To show that a CZO maps atoms into molecules we start with the following result
from [FJW]:
Theorem 5.14 ([FJW], Theorem 8.13) Let 0 < ε ≤ 1 and 0 < α < 1. If
T ∈ CZO(ε) ∩ WBP and T1 = 0, then T maps any smooth 0-atom aQ into a
fixed multiple of a smooth (ε, n + ε,−1)-molecule mQ .
Thus, if aQ is a smooth 0-atom for Q , then TaQ = cmQ , where mQ satisfies
1. |mQ(x)| ≤ |Q|−1/2
(1 +
|x− xQ|l(Q)
)−(n+ε)
,
65
2. |mQ(x)−mQ(y)| ≤ |Q|− 12
( |x− y|l(Q)
)ε
sup|z|≤|x−y|
(1 +
|x− z − xQ|l(Q)
)−(n+ε)
,
and c is uniform for all Q . Moreover, an (ε, n+ ε,−1)-molecule is a smooth molecule
for Bαqp (W ) (see Section 4.1) if 1 ≤ p < ∞ , 0 < q ≤ ∞ , 0 < α < ε and β < n + pα :
(i) δ = ε and 0 < α < ε ≤ 1,
(ii) J = np′ + β
p< n + α < n + ε = M ,
(iii) J − n− α = β−np− α < 0 =⇒ N = max([J − n− α],−1) = −1.
The next theorem follows by combining the two statements mentioned above, and
gives the boundedness of certain Calderon-Zygmund operators on Bαqp (W ) with some
restriction on the weight W :
Theorem 5.15 Suppose 0 < ε ≤ 1, 0 < α < ε, 1 ≤ p < ∞, 0 < q < ∞, and let
W satisfies any of (A1)-(A3). Assume β < n + pα. If T ∈ CZO(ε) ∩ WBP and
T1 = 0, then T extends to a bounded operator on Bαqp (W ).
Remark 5.16 If also T ∗1 = 0 in Theorem 5.14, then α = 0 can be included in the
range, since∫
Ta(x) dx = 〈Ta, 1〉 = 〈a, T ∗1〉 = 0 and so T maps any smooth 0-atom
into a smooth (ε, n + ε, 0)-molecule (see also [FJW, Cor. 8.21]).
Corollary 5.17 Let 1 ≤ p < ∞, 0 < q < ∞, 0 < ε ≤ 1 and 0 ≤ α < ε. Assume
β < n+pε. If T ∈ CZO(ε)∩WBP and T1 = T ∗1 = 0, then T extends to a bounded
operator on Bαqp (W ), in particular, for α = 0.
Proof. Since N = 0, the bound on β from the previous theorem can be relaxed
to β < n + pε . ¥
66
Remark 5.18 The condition T ∗(yγ) = 0 for |γ| ≤ N,N ≥ 1, produces more van-
ishing moments of a molecule Ta, so it is not difficult to satisfy (M.iii). But (M.ii)
M = n + ε > J = n + β−np
⇐⇒ β < n + pε creates a major restriction on the dou-
bling exponent of W . Note that in this case, we get that T maps any smooth 0-atom
into a smooth (ε, n + ε,N)-molecule, but this molecule is not a smooth molecule for
Bαqp (W ).
From now on N ≥ 0, since the case N = −1 is completely covered by Theorem 5.15.
Next we want to show that the restriction on the weight W (to be more precise the
restriction on the doubling exponent β ) can be removed in some cases by requiring
more smoothnes than (IIε ) on the kernel K .
We say that T ∈ CZO(N +ε), N ∈ N∪{0} , 0 < ε ≤ 1, if T is a continuous linear
operator from S(Rn) to S ′(Rn) and K , its distributional kernel defined on R2n�∆,
has the following properties:
(I) |K(x, y)| ≤ c
|x− y|n ,
(IIN ) |Dγ(2)K(x, y)| ≤ c
|x− y|n+|γ| , for |γ| ≤ N ,
(IIN+ε ) |Dγ(2)K(x, y)−Dγ
(2)K(x′, y)|+ |Dγ(2)K(y, x)−Dγ
(2)K(y, x′)|
≤ c|x− x′|ε
|x− y|n+|γ|+ε, for 2 |x− x′| ≤ |x− y| and |γ| = N ,
where the subscript 2 in Dγ(2) refers to differentiation with respect to the second
argument of K(x, y).
Note that CZO(ε) ⊇ CZO(N + ε) for N ≥ 0.
Theorem 5.19 Let 0 ≤ α < 1, 0 < ε ≤ 1, N0 ∈ N ∪ {0}. Suppose T ∈ CZO(N0 +
ε) ∩WBP , T1 = 0 and T ∗(yγ) = 0 for |γ| ≤ N0 . Then T maps any N0 -atom aQ
into a fixed multiple of a smooth (ε,N0 + n + ε,N0)-molecule.
67
More precisely, we will show that TaQ = cmQ with c independent of Q and
(i)
∫xγTaQ(x) dx = 0, for |γ| ≤ N0,
(ii) |TaQ(x)| ≤ c |Q|−1/2
(1 +
|x− xQ|l(Q)
)−(N0+n+ε)
,
(iii) |TaQ(x)−TaQ(y)| ≤ c |Q|− 12
[ |x− y|l(Q)
]ε
sup|z|≤|x−y|
(1 +
|z − (x− xQ)|l(Q)
)−(N0+n+ε)
.
Before we start the proof, we quote the following estimate, due to Meyer:
Lemma 5.20 ([M1]) Let T : D → D′ be a continuous linear operator with T ∈CZO(ε)∩WBP , 0 < ε ≤ 1 and T1 = 0. Then T maps D into L∞ and there exists
a constant c such that for any fixed z ∈ Rn , t > 0, ϕ ∈ D with supp ϕ ∈ Bt(z)
‖T ϕ‖L∞ ≤ c (‖ϕ‖L∞ + t ‖ 5 ϕ‖L∞).
Proof of Theorem 5.19. For simplicity, we give the proofs of (i), (ii) and
(iii) for Q = Q00 . The same methods apply to the general cube because of the
dilation-translation nature of the estimates. Thus, consider the unit atom a = aQ00
with xQ00 = 0 and l(Q00) = 1. First property (i) immediately follows from the fact
that T ∗(yγ) = 0 for |γ| ≤ N0 . To get (ii) we consider two cases: |x| ≤ 6√
n and
|x| > 6√
n . For |x| ≤ 6√
n , use Lemma 5.20 to obtain
|T a(x)| ≤ ‖T a‖L∞ ≤ c (‖a‖L∞ + ‖ 5 a‖L∞) ≤ c.
If |x| > 6√
n , we get
|Ta(x)| =∣∣∣∣∫
K(x, y) a(y) dy
∣∣∣∣
=
∣∣∣∣∣∣
∫
3Q00
K(x, y)−
∑
|γ|≤N0
Dγ(y)K(x, 0)
γ!yγ
a(y) dy
∣∣∣∣∣∣, (5.9)
68
since aQ is an N0 -atom, and thus, has N0 vanishing moments∫
yγaQ(y) dy = 0 for
|γ| ≤ N0 . Then (5.9) is bounded by
∫
3Q00
∑
|γ|=N0
∣∣∣∣[Dγ
(y)K(x, θ(y))−Dγ(y)K(x, 0)
] yγ
γ!
∣∣∣∣ |a(y)| dy.
Note that if y ∈ supp a , then 2 |θ(y)| ≤ 2 |y| ≤ 2 ·3√n < |x| , and, using the property
(IIN+ε ) of the kernel K to estimate the difference, we get
∑
|γ|=N0
|Dγ(y)K(x, θ(y))−Dγ
(y)K(x, 0)| ≤ c|θ(y)|ε|x|n+|γ|+ε
≤ c|y|ε
|x|n+N0+ε.
Thus,
|Ta(x)| ≤ cn,N0
|x|n+N0+ε
∫
3Q00
|y|N0+ε |a(y)| dy ≤ c
|x|n+N0+ε.
In order to show (iii), we prove that
|Ta(x)−Ta(x′)| ≤ c |x−x′|ε(
1
(1 + |x|)n+N0+ε+
1
(1 + |x′|)n+N0+ε
). (5.10)
In the case |x−x′| ≥ 1, the estimate (5.10) follows trivially from (ii) and the triangle
inequality. For |x− x′| < 1 and |x| > 10√
n , we use vanishing moments of a(x) and
the integral form of the remainder to get
|Ta(x)− Ta(x′)| =∣∣∣∣∫
(K(x, y)−K(x′, y)) a(y) dy
∣∣∣∣ =
∣∣∣∣∫
3Q00
[K(x, y)
−∑
|γ|≤N0−1
Dγ(y)K(x, 0)
γ!yγ −K(x′, y) +
∑
|γ|≤N0−1
Dγ(y)K(x′, 0)
γ!yγ
a(y) dy
∣∣∣∣∣∣
≤∫
3Q00
∫ 1
0
(1− s)N0−1
(N0 − 1)!
∑
|γ|=N0
∣∣∣Dγ(y)K(x, sy)−Dγ
(y)K(x′, sy)∣∣∣ |y|
γ
γ!|a(y)| ds dy.
If |x| ≥ 10√
n and y ∈ supp a , then |x− sy| ≥ |x|− s|y| ≥ 10√
n− 3√
n ≥ 2 |x−x′|and also |x − sy| ≥ |x| − s|y| ≥ |x| − 3
√n ≥ |x| − |x|
2≥ |x|
2. By (IIN+ε ) the last
integral is bounded by
c
∫
3Q00
∫ 1
0
|x− x′|ε|x− sy|n+N0+ε
|y|N0 ds dy ≤ c|x− x′|ε|x|n+N0+ε
.
69
In case |x − x′| < 1 and |x| ≤ 10√
n , an exact repetition of the argument on p. 85
of [FJW] or part (c) on p.62 of [FTW] shows that
|Ta(x)− Ta(x′)| ≤ c |x− x′|ε
by using the decay property (I) and the Lipschitz condition (II0+ε ) of the kernel K ,
which holds for any CZO(N0 + ε), N0 ≥ 0. This completes the proof. ¥
Corollary 5.21 Let 1 ≤ p < ∞, 0 < q < ∞, and let W satisfy any of (A1)-
(A3). Suppose 0 ≤ α ≤ β−np− [β−n
p], where β is the doubling exponent of W . Let
N = [β−np− α] and β−n
p− [β−n
p] < ε ≤ 1. If T ∈ CZO(N + ε) ∩WBP , T1 = 0 and
T ∗(yγ) = 0 for |γ| ≤ N , then T extends to a bounded operator on Bαqp (W ).
Proof. By the previous theorem T maps any smooth N -atom into a smooth
(ε,N + n + ε,N)-molecule. This molecule is a smooth molecule for Bαqp (W ) if
(i) α < ε ≤ 1,
(ii) M = N + n + ε > J = n + β−np
⇐⇒ [β−np− α] = [β−n
p] > β−n
p− ε and
(iii) N = max([J − n− α],−1) = [β−np− α] , which are all true.
By the boundedness criterion (Lemma 5.13), T is bounded on Bαqp (W ). ¥
Corollary 5.22 Let 1 ≤ p < ∞, 0 < q < ∞, and let W satisfy any of (A1)-(A3).
Suppose 0 ≤ β−np− [β−n
p] < α < 1, where β is the doubling exponent of W . Let
N = [β−np− α] and α < ε ≤ 1. If T ∈ CZO(N + 1 + ε) ∩ WBP , T1 = 0 and
T ∗(yγ) = 0 for |γ| ≤ N + 1, then T is bounded on Bαqp (W ).
Proof. By Theorem 5.19, T maps any smooth (N + 1)-atom into a smooth
(ε,N + 1 + n + ε,N + 1)-molecule, which is also a smooth (ε,N + 1 + n + ε, N)-
molecule. This one, in its turn, is a smooth molecule for Bαqp (W ), since
(i) α < ε ≤ 1,
70
(ii) M = N + 1 + n + ε > J = n + β−np
⇐⇒ [β−np− α] + 1 > β−n
p− ε and
(iii) N = max([J − n− α],−1) = [β−np− α] .
By the boundedness criterion (Lemma 5.13), T extends to a bounded operator on
Bαqp (W ). ¥
Remark 5.23 Note that the condition T ∗(yγ) = 0, |γ| ≤ N , can be very restrictive;
for example, the Hilbert transform does not satisfy this condition for |γ| > 0. On the
other hand, we have considered a general class of CZOs, not necessarily of convolution
type. Utilizing the convolution structure will let us drop the above condition.
Let N ∈ N ∪ {0} . Let T be a convolution operator, i.e., the kernel K(x, y) =
K(x− y) is defined on Rn�{0} and satisfies
(C.1) |K(x)| ≤ c
|x|n ,
(C.2) |DγK(x)| ≤ c
|x|n+|γ| , for |γ| ≤ N + 1,
(C.3)
∫
R1<|x|<R2
K(x) dx = 0, for all 0 < R1 < R2 < ∞ .
Remark 5.24 We replace (IIN ) and (IIN+ε ) of the general CZO kernel with the
slightly stronger smoothness condition (C.2) to make the proof below more concise.
The reader can check that conditions corresponding to (IIN ) and (IIN+ε ) in the con-
volution case would suffice for the statements below.
Now we obtain an analog of Theorem 5.19 saying that T maps smooth atoms into
smooth molecules, and then we show the boundedness of T .
Theorem 5.25 Let 0 ≤ α < 1, 0 < ε ≤ 1, N ∈ N ∪ {0}. Let T be a convolution
operator with a kernel K satisfying (C.1)-(C.3). Then T maps any smooth N -atom
aQ into a fixed multiple of a smooth (ε, N + 1 + n,N)-molecule.
71
More precisely, we will show that
(i)
∫xγTaQ(x) dx = 0, for |γ| ≤ N, and
(ii) |DγTaQ(x)| ≤ c |Q|−1/2−|γ|/n
(1 +
|x− xQ|l(Q)
)−(N+n+1)
for |γ| = 0, 1.
By the Mean Value Theorem, (ii) with |γ| = 1 implies the Lipschitz condition (M4)
for |γ| = 0.
Proof. To obtain (ii) we first consider x /∈ 10√
n Q . Then
|TaQ(x)|, | 5 TaQ(x)| =∣∣∣∣∣∣
∑
|γ0|=0 or 1
∫
3Q
Dγ0K(x− y) aQ(y) dy
∣∣∣∣∣∣
=
∣∣∣∣∣∣∑
|γ0|=0 or 1
∫
3Q
[Dγ0K(x− y)
−∑
|γ|≤N−|γ0|
DγDγ0K(x− xQ)
γ!(xQ − y)γ
aQ(y) dy
∣∣∣∣∣∣(5.11)
since aQ is an N -atom, and thus, has N vanishing moments∫
yγaQ(y) dy = 0 for
|γ| ≤ N . Then (5.11) is bounded by
∑
|γ0|=0 or 1
∫
3Q
∑
|γ|=N+1−|γ0|
|Dγ+γ0K(x− xQ + θ(y − xQ))|γ!
|xQ − y|N−|γ0|+1 |aQ(y)| dy,
for some 0 ≤ θ ≤ 1. Since x /∈ 10√
nQ and y ∈ 3Q , |y−xQ| ≤ 2√
n l(Q) ≤ 12|x−xQ| .
Using property (C.2) of the kernel K , we get
|DγK(x− xQ + θ(y − xQ))| ≤ c
|x− xQ + θ(y − xQ)|n+|γ| ≈c
|x− xQ|n+|γ| .
So,
|TaQ(x)|, | 5 TaQ(x)| ≤ cn,N
|x− xQ|n+N+1
∫
3Q
|xQ − y|N−|γ0|+1 |aQ(y)| dy
≤ c[l(Q)]N−|γ0|+1
|x− xQ|n+N+1|Q|−1/2 |Q| = c |Q|−1/2−|γ0|/n
[l(Q)
|x− xQ|]n+N+1
,
by the properties of aQ .
72
If x ∈ 10√
n Q and y ∈ 3Q , then |x − y| ≤ 13nl(Q), so by the cancellation
property (C.3) of K (using Dγ(K ∗ aQ) = K ∗ (DγaQ)), we obtain
|TaQ(x)|, | 5 TaQ(x)| ≤∑
|γ0|=0 or 1
∣∣∣∣∫
3Q
K(x− y) Dγ0aQ(y) dy
∣∣∣∣
=∑
|γ0|=0 or 1
∣∣∣∣∫
3Q
K(x− y) [Dγ0aQ(y)−Dγ0aQ(x)] dy
∣∣∣∣
≤ c
∫
|y−x|≤13nl(Q)
1
|x− y|n |Q|−1/2−|γ0|/n−1/n |x− y| dy
≤ c |Q|−1/2−|γ0|/n−1/n
∫ 13nl(Q)
0
r−n r rn−1 dr = c |Q|−1/2−|γ0|/n.
This concludes the proof of (ii).
Property (i) comes from the fact that T is a convolution operator and aQ has
vanishing moments up to order N . Property (ii) guarantees the absolute convergence
of the integral in (i). ¥
Corollary 5.26 Convolution operators with kernels satisfying (C.1)-(C.3) are
bounded on Bαqp (W ) if W satisfies any of (A1)-(A3) and 0 ≤ α < ε ≤ 1, 0 < q < ∞,
1 < p < ∞. In particular, the Hilbert transform H (n = 1) is bounded on Bαqp (W )
and the Riesz transforms Rj , j = 1, ..., n (n ≥ 2), are bounded on Bαqp (W ).
Proof. This is an immediate consequence of Theorem 5.25 and Lemma 5.13:
choose N = [β−np− α] in Theorem 5.25; then T maps any smooth N -atom into a
smooth (ε,N +1+n,N)-molecule, which is either a smooth (ε,N +1+n,N)-molecule
for Bαqp (W ), if α ≤ β−n
p− [β−n
p] or an (ε,N + 1 + n,N − 1)-molecule for Bαq
p (W ), if
1 > α > β−np− [β−n
p] . Note that both Hilbert and Riesz transforms are convolution
type operators with kernels satisfying (C.1)-(C.3). ¥
73
CHAPTER 6
Application to Wavelets
Consider a pair (ϕ, ψ) from A with the mutual property (2.1). Then the family
{ϕQ, ψQ} behaves similarly to an orthonormal system because of the property
f =∑Q
〈f, ϕQ〉ψQ =∑Q
sQ ψQ for all f ∈ S ′/P .
However, this system does not constitute an orthonormal basis. This can be achieved
by the Meyer and Lemarie construction of a wavelet basis with the generating function
θ ∈ S (see [LM] and [M1]):
Theorem 6.1 There exist real-valued functions θ(i) ∈ S(Rn), i = 1, ..., 2n − 1, such
that the collection {θ(i)νk} = {2νn/2θ(i)(2νx − k)} is an orthonormal basis for L2(Rn).
The functions θ(i) satisfy
supp θ(i) ⊆{[−8
3π,
8
3π
]n
\[−2
3π,
2
3π
]n}
and, hence, ∫
Rxγθ(x) dx = 0 for all multi-indices γ.
Thus, we have f =2n−1∑i=1
∑Q
⟨f, θ
(i)Q
⟩θ
(i)Q for all f ∈ L2(Rn). This identity extends to
all f ∈ S ′/P(Rn).
74
Theorem 6.2 Let α ∈ R, 0 < q ≤ ∞, 1 ≤ p < ∞, and let W satisfy any of (A1)-
(A3). Let θ(i) , i = 1, ..., 2n − 1, be generating wavelet functions as in Theorem 6.1.
Then∥∥∥~f
∥∥∥Bαq
p (W )≈
2n−1∑i=1
∥∥∥∥{⟨
~f, θ(i)Q
⟩}Q
∥∥∥∥bαqp (W )
.
Proof. Assume i = 1, ..., 2n − 1. Since {θ(i)Q }Q,i is a family of smooth molecules
for Bαqp (W ), the inequality
∥∥∥~f∥∥∥
Bαqp (W )
≡∥∥∥∥∥∑Q,i
⟨~f, θ
(i)Q
⟩θ
(i)Q
∥∥∥∥∥Bαq
p (W )
≤ c∑
i
‖{⟨
~f, θ(i)Q
⟩}Q‖bαq
p (W ) (6.1)
follows immediately from Theorem 4.2. Therefore, we need to focus only on the
opposite direction.
Let ϕ ∈ A be such that∑
ν∈Z |ϕ(2νξ)|2 = 1 for ξ 6= 0. Let ~sQ =⟨
~f, ϕQ
⟩.
Applying the boundedness of the ϕ-transform (Theorem 4.15), we obtain
∥∥∥{~sQ}Q
∥∥∥bαqp (W )
≤ c∥∥∥~f
∥∥∥Bαq
p (W ). (6.2)
Now for each i and Q , define ~t(i)
Q =⟨
~f, θ(i)Q
⟩. Since θ
(i)Q ∈ S , by the ϕ-transform
decomposition with ψ = ϕ , we have θ(i)Q =
∑P
⟨θ
(i)Q , ϕP
⟩ϕP , which gives
~t(i)
Q =∑
P
⟨θ
(i)Q , ϕP
⟩ ⟨~f, ϕP
⟩=
∑P
a(i)QP ~sP .
Since supp ϕP ∩ supp θ(i)Q 6= {∅} only if l(Q) = 2jl(P ) with j = 1, 2, 3, 4 (recall
that supp ϕP ⊆ {ξ ∈ Rn : 2µ−1 ≤ |ξ| ≤ 2µ+1} when l(P ) = 2−µ ), we see that
a(i)QP =
⟨θ
(i)Q , ϕP
⟩= 0 unless 2 ≤ l(Q)
l(P )≤ 16, in which case
|aQP | ≤ cM
(1 +
|xQ − xP |l(Q)
)−M
for each M > 0,
as was shown in [FJW], p. 72. Let M > np′ + β
p. Then A(i) :=
(a
(i)QP
)is an almost
diagonal matrix for each i , and, by Theorem 1.10,
‖{~t (i)Q }Q‖bαq
p (W ) ≤ c ‖{~sQ}Q‖bαqp (W ). (6.3)
75
Combining (6.3) with (6.2) we get the opposite direction of (6.1). ¥
Corollary 6.3 Let {Nψ(i)}, i = 1, ..., 2n − 1, be a collection of Daubechies DN
generating wavelet functions for L2(Rn) with compact supports linearly dependent on
N (for more details, see [D]). Then for any ~f with fj ∈ S ′/P(Rn), j = 1, ...,m,
∥∥∥~f∥∥∥
Bαqp (W )
≈2n−1∑i=1
∥∥∥∥{⟨
~f, Nψ(i)Q
⟩}Q
∥∥∥∥bαqp (W )
(6.4)
for sufficiently large N .
Proof. First, observe that there exists a constant c such that for all i = 1, ..., 2n−1, the functions
Nψ(i)
care smooth molecules, and so
{Nψ
(i)Q
c
}
Q
is a family of
smooth molecules for Bαqp (W ) if we choose N sufficiently large to have the necessary
smoothness and vanishing moments. Second, if ϕ ∈ A , then
(⟨Nψ
(i)Q , ϕP
⟩QP
)∈
adαqp (β) by (5.2). Applying these two facts in the proof of the previous theorem, we
get (6.4). ¥
76
CHAPTER 7
Duality
7.1 General facts on duality
An important tool that we need is the duality on lq(X) with X being a Banach space.
By definition lq(X), 0 < q < ∞ is the set of all sequences {fν}ν∈Z with fν ∈ X ,
ν ∈ Z such that
(∑
ν∈Z‖fν‖q
X
)1/q
< ∞ . If 1 ≤ q < ∞ , then (lq(X))∗ = lq′(X∗) (see
[D, Chapter 8]), and if g is a continuous linear functional on lq(X) identified with
{gν}ν∈Z ∈ lq′(X∗), then the duality is represented as
g(f) = (f, g) =∑
ν∈Z〈fν , gν〉X ,
where 〈fν , gν〉X = gν(fν) is the pairing between X and X∗ . We will mainly be
concerned with X = Lp , 1 ≤ p < ∞ , or Lp(W ), 1 < p < ∞ , and, thus, X∗ = Lp′ or
Lp′(W−p′/p), respectively, with the pairing 〈f, g〉X =∫ 〈f(x), g(x)〉H dx .
If 0 < q < 1, and X = Lp , 1 ≤ p < ∞ , then (lq(Lp))∗ = l∞(Lp′) (see [T, p.177])
and the pairing is defined as above.
7.2 Duality of sequence Besov spaces
Theorem 7.1 Let W be a matrix weight, α ∈ R, 0 < q < ∞, 1 < p < ∞. Then
77
(i) b−αq′p′ (W−p′/p) ⊆
[bαqp (W )
]∗always
(ii)[bαqp (W )
]∗⊆ b−αq′
p′ (W−p′/p) if W ∈ Ap .
We will prove this theorem, which implies (1.12) of Theorem 1.17, in several steps.
The use of reducing operators is essential and helps to understand why certain con-
ditions on the weight W are necessary.
Proof of (i) of Theorem 7.1. For each ~t ∈ b−αq′p′ (W−p′/p) define a functional
lt on bαqp (W ) by
lt(~s ) = (~s,~t ) =∑
Q
⟨~sQ,~tQ
⟩H for any ~s = {~sQ}Q ∈ bαq
p (W ).
The calculations below show that this sum converges and lt ∈[bαqp (W )
]∗:
∣∣∣∣∣∑
Q
⟨~sQ,~tQ
⟩H
∣∣∣∣∣ ≤∑
ν∈Z
∑Q∈Qν
| ⟨~sQ,~tQ⟩H | =
∑
ν∈Z
∫
Rn
∑Q∈Qν
∣∣⟨~sQ,~tQ⟩H∣∣ |Q|−1χQ(t) dt
∑
ν∈Z
∫
Rn
∑Q∈Qν
|Q|−1∣∣⟨W−1/p(x) W 1/p(x)~sQ,~tQ
⟩H∣∣ χQ(x) dx. (7.1)
Using the self-adjointness of W and the Cauchy-Schwarz inequality, we bound (7.1)
by
∑
ν∈Z
∫
Rn
∑Q∈Qν
(|Q|−α
n− 1
2 χQ(x) ‖W 1/p(x)~sQ‖H)(|Q|αn− 1
2 χQ(x) ‖W−1/p(x)~tQ‖H)
dx.
Applying Holder’s inequality several times, we estimate lt(~s ) by
∑
ν∈Z
∫
Rn
( ∑Q∈Qν
(|Q|−α
n− 1
2 χQ(x) ‖W 1/p(x)~sQ‖H)p
) 1p
×( ∑
Q∈Qν
(|Q|αn− 1
2 χQ(x) ‖W−1/p(x)~tQ‖H)p′
) 1p′
dx
≤∑
ν∈Z
∥∥∥∥∥∑
Q∈Qν
|Q|−αn− 1
2 χQ~sQ
∥∥∥∥∥Lp(W )
∥∥∥∥∥∑
Q∈Qν
|Q|αn− 12 χQ
~tQ
∥∥∥∥∥Lp′ (W−p′/p)
(7.2)
78
≤ ‖~s ‖bαqp (W ) ‖~t ‖b−αq′
p′ (W−p′/p),
for 1 < q < ∞ . In case of 0 < q ≤ 1, we bound (7.2) by ‖~s ‖bα1p (W ) ‖~t ‖b−α∞
p′ (W−p′/p) .
Since lq is embedded into l1 when 0 < q ≤ 1, we estimate the previous product by
‖~s ‖bαqp (W ) ‖~t ‖b−α∞
p′ (W−p′/p) . ¥
In terms of reducing operators (or using (1.10)) the previous lemma states
b−αq′p′ ({A#
Q}) ⊆[bαqp ({AQ})
]∗. (7.3)
If we follow along the lines of the proof again but instead of W−1/p(t) W 1/p(t) in (7.1)
use A−1Q AQ , then we obtain the following statement.
Lemma 7.2 Let α ∈ R, 0 < q < ∞, 1 ≤ p < ∞ and {AQ}Q ∈ RSD . Then
b−αq′p′ ({A−1
Q }) ⊆[bαqp ({AQ})
]∗. (7.4)
In fact, if we have only proven (7.4), then (7.3) (and equivalently part (i) of Theorem
7.1) could have been obtained as a consequence of (7.4) and Corollary 4.20, i.e.,
b−αq′p′ ({A#
Q}) ⊆ b−αq′p′ ({A−1
Q }) ⊆[bαqp ({AQ})
]∗. (7.5)
Observe that (7.4) holds for any {AQ}Q ∈ RSD , not necessarily generated by W .
Now we will study the opposite embeddings. By Lemma 7.3 below, we will get
[bαqp ({AQ})
]∗⊆ b−αq′
p′ ({A−1Q }) (7.6)
without any additional assumptions on the sequence {AQ}Q . Note that combining
(7.4) and (7.6), we obtain (1.13). Applying Corollary 4.20 again, (7.6) is continued
as[bαqp ({AQ})
]∗⊆ b−αq′
p′ ({A−1Q })
Ap⊆ b−αq′p′ ({A#
Q}) ≈ b−αq′p′ (W−p′/p)
with the second embedding being held under Ap condition. Thus, the embedding (ii)
of Theorem 7.1 holds if W ∈ Ap .
79
Lemma 7.3 Let {AQ}Q ∈ RSD , α ∈ R, 0 < q < ∞, 1 ≤ p < ∞. Then (7.6) holds.
Proof. Let l ∈[bαqp ({AQ})
]∗. We show that there exists ~t ∈ b−αq′
p′ ({A−1Q }) such
that for any ~s ∈ bαqp ({AQ})
l(~s ) = (~s,~t ) =∑Q
⟨~sQ,~tQ
⟩H and ‖~t ‖
b−αq′p′ ({A−1
Q }) ≤ ‖l‖.
Let ~e(k)J denote a vector-valued sequence enumerated by dyadic cubes such that in
the kth component (kth row) of this vector the J th entry (corresponding to the dyadic
cube J ) is equal to 1 and all other entries are zero:
~e(k)J = (..., {0}Q, ..., {...0...1Jthentry...0...}Q − kthrow, ..., {0}Q, ...)T .
Now if ~s has only finitely many non-zero entries, i.e.,
~s =∑
Q finite
m∑
k=1
s(k)Q ~e
(k)Q ,
then by linearity
l(~s ) =∑
Q finite
m∑
k=1
s(k)Q l(~e
(k)Q ) =:
∑
Q finite
m∑
k=1
s(k)Q t
(k)Q .
By continuity, since finitely non-zero sequences are dense (p, q < ∞), we get
l(~s ) =∑Q∈D
m∑
k=1
s(k)Q t
(k)Q =
∑Q∈D
⟨~sQ,~tQ
⟩H for any ~s ∈ bαq
p ({AQ}).
Now everything is set up to show that ~t : = ({t(1)Q }Q, {t(2)
Q }Q, ..., {t(m)Q }Q)T ∈
b−αq′p′ ({A−1
Q }). For ~s ∈ bαqp (Rm), set ~sQ = A−1
Q~s and define
l(~s ) := l({A−1Q
~sQ}Q) = l({~sQ}Q) =∑Q
⟨~sQ,~tQ
⟩H =
∑Q
⟨AQ~sQ, A−1
Q~tQ
⟩H
=∑
Q
⟨~sQ, ~tQ
⟩H
,
where ~tQ = A−1Q
~tQ . By above,
| l(~s )| ≤ c ‖{~sQ}Q‖bαqp ({AQ}) = c ‖{~sQ}Q‖bαq
p (Rm),
80
i.e., l induces a continuous linear functional l on bαqp (Rm). By Lemma 7.4 below
{~tQ}Q ∈ b−αq′p′ (Rm). Since the inside Lp′ -norm of the b−αq′
p′ (Rm)-norm of ~t is
∥∥∥∥∥∑
Q∈Qν
|Q|− 12~tQχQ
∥∥∥∥∥Lp′
=
∥∥∥∥∥∑
Q∈Qν
|Q|− 12‖A−1
Q~tQ‖HχQ
∥∥∥∥∥Lp′
=
∥∥∥∥∥∑
Q∈Qν
|Q|− 12~tQχQ
∥∥∥∥∥Lp′ ({A−1
Q })
,
~t ∈ b−αq′p′ ({A−1
Q }) and the lemma is proved. ¥
Lemma 7.4 Let α ∈ R, 0 < q < ∞, 1 ≤ p < ∞. Then
[bαqp (Rm)
]∗≈ b−αq′
p′ (Rm). (7.7)
Proof. It suffices to show only the scalar case (m = 1) of (7.7), since ~s ∈ bαqp (Rm)
means that each component s(i) belongs to bαqp and by making zero all but one of the
components of an arbitrary ~s we obtain (7.7).
The embedding[bαqp
]∗⊇ b−αq′
p′ is a trivial application of Holder’s inequality plus
the embedding bαqp → bα1
p for q < 1, so we concentrate only on the opposite embed-
ding.
Suppose l ∈[bαqp
]∗. Using linearity and continuity, l can be represented by some
sequence {tQ}Q as l(s) =∑
Q sQtQ for any s = {sQ} ∈ bαqp and
|l(s)| =∣∣∣∣∣∑Q
sQtQ
∣∣∣∣∣ ≤ ‖l‖ ‖s‖bαqp
. (7.8)
Case q ≥ 1: For each ν ∈ Z let fν(s)(x) =∑
Q∈Qν
|Q|−αn− 1
2 sQχQ(x). Define a map
I : bαqp −→ lq(Lp) by I(s) = {fν(s)}ν∈Z . Observe that ‖I(s)‖lq(Lp) = ‖s‖bαq
p, in
other words, by the natural construction I is a linear isometry onto the subspace
I(bαqp ) of lq(Lp). Then l induces a continuous linear functional l on I(bαq
p ) ⊆ lq(Lp)
(continuous in lq(Lp)-norm) by l(I(s)) = l(s). Since lq(Lp) is a Banach space, by
the Hahn-Banach Theorem l extends to a continuous linear functional lext on all of
lq(Lp) by lext |I(bαqp )= l with ‖lext‖ = ‖l‖ ≤ ‖l‖ . Since [lq(Lp)]∗ = lq
′(Lp′), lext is
81
represented by a sequence g = {gν}ν∈Z ∈ lq′(Lp′) with ‖g‖ = ‖{gν}ν‖lq′ (Lp′ ) ≤ ‖l‖
and
∑Q
sQtQ = l(s) = l({fν(s)}) =∑
ν∈Z
∫
Rn
fν(s)(x)gν(x) dx, for any ~s ∈ bαqp ,
or∑Q
sQtQ =∑
ν∈Z
∑Q∈Qν
|Q|−αn− 1
2 sQ
∫
Q
gν(x) dx.
Taking sQ = 0 for all but one cube, we get tQ = |Q|−αn
+ 12 < gν >Q . Using Holder’s
inequality, we have
‖t‖b−αq′p′
=
∥∥∥∥∥∥
∥∥∥∥∥∑
Q∈Qν
< gν >Q χQ
∥∥∥∥∥Lp′
ν
∥∥∥∥∥∥lq′
≤ ‖{~gν}ν‖lq′(Lp′ ) ≤ ‖l‖.
Case 0 < q < 1: Suppose 1 < p < ∞ . Fix ν ∈ Z and let Fν denote a finite
collection of cubes from Qν . Set τν =∑
Q∈Fν
(|Q|αn− 1
2+ 1
p′ |tQ|)p′
. Since the sum is
finite, τν < ∞ . Let sQ = |Q|(αn− 1
2+ 1
p′ )p′ |tQ|p′−2 tQ , if Q ∈ Fν and tQ 6= 0; otherwise
let sQ = 0. Note that ‖{sQ}Q‖bαqp
= τ1/pν . Observe that
∑Q sQtQ = τν and by (7.8)
τν ≤ ‖l‖ ‖s‖bαqp
= ‖l‖ τ1/pν . Since τν is finite, we get τ
1/p′ν ≤ ‖l‖ and the estimate
holds independently of the collection Fν taken. Hence, we can pass to the limit from
Fν to Qν . Then,
‖t‖b−α∞p′
= supν∈Z
( ∑Q∈Qν
(|Q|αn− 1
2+ 1
p′ |tQ|)p′
)1/p′
= supν∈Z
τ 1/p′ν ≤ ‖l‖ or t ∈ b−α∞
p′ .
Now assume p = 1. Fix P ∈ D and set s(P ) ={
s(P )Q
}Q
by s(P )Q = |Q|αn− 1
2 sgn tQ
if Q = P and s(P )Q = 0 otherwise. Then
∥∥∥∥{
s(P )Q
}Q
∥∥∥∥bαq1
= 1 and |P |αn− 12 |tP | =
∑Q
s(P )Q tQ = l
(s(P )
) ≤ ‖l‖∥∥∥∥{
s(P )Q
}Q
∥∥∥∥bαq1
= ‖l‖ for any P ∈ D . Hence,
‖t‖b−α∞∞ = supP∈D
|P |αn− 12 |tP | ≤ ‖l‖ or t ∈ b−α∞
∞ .
¥
82
7.3 Equivalence of sequence and discrete averag-
ing Besov spaces
In this section we discuss norm equivalence between Bαqp ({AQ}) and bαq
p ({AQ}). We
suppose α ∈ R , 0 < q ≤ ∞ and 1 ≤ p < ∞ for all statements in this section. If
q = ∞ , then set q′ = 1.
Lemma 7.5 If {AQ} is a doubling sequence of order p, then for ~sQ =⟨
~f, ϕQ
⟩
‖{~sQ}Q‖bαqp ({AQ}) ≤ c ‖~f ‖Bαq
p ({AQ}). (7.9)
Proof. Note that ~sQ = |Q|1/2(ϕν ∗ ~f )(2−νk) for Q = Qνk , where ϕ(x) = ϕ(−x).
Let ‖{~sQ}Q‖bαqp ({AQ}) =:
∥∥{J1/p
ν
}ν
∥∥lαq
, where
Jν =∑
k∈Zn
∫
Qνk
‖AQνk(ϕν ∗ ~f )(2−νk)‖p dx. (7.10)
Since ϕν ∗ ~f ∈ Eν , Lemma 4.9 implies
(ϕν ∗ ~f )(2−νk) =∑
l∈Zn
(ϕν ∗ ~f )(2−νl + x) γ(k − l − 2νx), x ∈ Qνk
for some γ ∈ Γ. Then
Jν ≤∑
k∈Zn
∫
Qνk
(∑
l∈Zn
‖AQνk(ϕν ∗ ~f )(2−νl + x)‖ |γ(k − l − 2νx)|
)p
dx
≤ c∑
k∈Zn
∫
Qνk
(∑
l∈Zn
‖AQνk(ϕν ∗ ~f )(2−νl + x)‖
(1 + |k − l − 2νx|M)p
dx for some M > β + n.
Using the discrete Holder inequality and the fact that M > n , we bring the pth power
inside the sum on l . Furthermore, since {AQ}Q is doubling, (1.7) implies
‖AQνk~u ‖p ≤ c (1 + |l|)β ‖AQν(k+l)
~u ‖p, for any ~u ∈ H. (7.11)
Thus,
Jν ≤ c∑
k∈Zn
∫
Qνk
∑
l∈Zn
(1 + |l|)β‖AQν(k+l)(ϕν ∗ ~f )(2−νl + x)‖p
(1 + |k − l − 2νx|)Mdx.
83
Changing variable (t = x + 2−νl) and reindexing the sum on l , we get
Jν ≤ c∑
k∈Zn
∫
Qνl
∑
l∈Zn
(1 + |k − l|)β−M‖AQνl(ϕν ∗ ~f )(t)‖p dt
≤ c∑
l∈Zn
∫
Qνl
‖AQνl(ϕν ∗ ~f )(t)‖p dt = c ‖ϕν ∗ ~f ‖Lp({AQ},ν)
(the sum on k converges since M − β > n). Thus,
‖{~sQ}Q‖bαqp ({AQ}) ≤ c ‖~f ‖Bαq
p ({AQ},ϕ).
Now we need an independence of the space Bαqp ({AQ}) on the choice of ϕ (or ϕ). We
apply the same strategy as in Theorem 4.15, namely, we use the proof of Corollary
7.10 below, which will imply that the last expression is equivalent to c ‖~f ‖Bαqp ({AQ},ϕ)
and, thus, (7.9) is proved. ¥
Corollary 7.6 If {AQ} is a doubling sequence of order p, then for ~sQ =⟨
~f, ϕQ
⟩
‖{~sQ}Q‖bαqp ({A−1
Q }) ≤ c ‖~f ‖Bαqp ({A−1
Q }). (7.12)
and
‖{~sQ}Q‖b−αq′p′ ({A−1
Q }) ≤ c ‖~f ‖B−αq′
p′ ({A−1Q }). (7.13)
Proof. For (7.12) repeat the previous proof with each AQ replaced by A−1Q and
instead of the estimate (7.11) use
‖A−1Qνk
~u ‖p ≤ c (1 + |l|)β ‖A−1Qν(k+l)
~u ‖p, for any u ∈ H, (7.14)
which follows from the doubling property (1.7) and duality ‖A−1Q ~u ‖ = sup
~v 6=0
|(~u,~v )|‖AQ~v ‖ .
For (7.13) use the obvious replacements for α , p , q and AQ . If 1 < p < ∞ , choose
M > βp′/p + n and replace (7.11) by
‖A−1Qνk
~u ‖p′ ≤ c (1 + |l|)βp′/p ‖A−1Qν(k+l)
~u ‖p′ , for any ~u ∈ H, (7.15)
84
which is obtained from (7.14) by raising to the power p′/p . If p = 1 (p′ = ∞), then
replace (7.10) with the L∞ -norm:
Jν = supx∈Rn
∑
k∈Zn
‖A−1Qνk
(ϕν ∗ ~f )(2−νk)‖χQνk(x)
and use (7.14) instead of (7.11) to get
Jν ≤ c supt∈Rn
∑
l∈Zn
‖A−1Qνl
(ϕν ∗ ~f )(t)‖χQνl(t) = c ‖ϕν ∗ ~f ‖L∞({A−1
Q },ν).
¥
Lemma 7.7 Suppose {AQ}Q is a doubling sequence of order p. Then
‖~f ‖Bαqp ({AQ}) ≤ c
∥∥∥∥{~sQ(~f )
}Q
∥∥∥∥bαqp ({AQ})
. (7.16)
Proof. Using ~f =∑
Q
~sQ(~f )ψQ , we get
∥∥∥∥∥∑
Q
~sQ(~f )ψQ
∥∥∥∥∥Bαq
p ({AQ})
≤
∥∥∥∥∥∥∥
∑
µ∈Z
∑
l(P )=2−ν
∫
P
∑
l(Q)=2−µ
‖AP~sQ‖|(ϕν ∗ ψQ)(x)|
p
dx
1/p
ν
∥∥∥∥∥∥∥lαq
=
∥∥∥∥∥∥∥
ν+1∑µ=ν−1
∑
l(P )=2−ν
∫
P
∑
l(Q)=2−µ
‖AP~sQ‖|(ϕν ∗ ψQ)(x)|
p
dx
1/p
ν
∥∥∥∥∥∥∥lαq
=:∥∥{
J1/pν
}ν
∥∥lαq
,
since ϕν ∗ ψQ = 0 if |µ− ν| > 1. Using the convolution estimates (4.2) and (4.3), we
get (for any M > 0)
|(ϕν∗ψQ)(x)| ≤ cM |Q|−1/2(1+2ν |x−xQ|)−M , if µ = ν−1, ν, ν+1. (7.17)
85
If 1 < p < ∞ , choose M = M1 + M2 with M1 > β/p + n/p and M2 > n/p′ ;
if p = 1, let M = M1 > β + n . Then applying the above estimate and Holder’s
inequality, we obtain
Jν ≤ c
ν+1∑µ=ν−1
∑
l(P )=2−ν
∑
l(Q)=2−µ
‖AP~sQ‖p |P | |Q|−p/2 (1 + 2ν |xP − xQ|)−M1p.
Shifting AP to AQ by doubling, we get
Jν ≤ c
ν+1∑µ=ν−1
∑
l(Q)=2−µ
|Q|−p/2 ‖AQ~sQ‖p |Q|∑
l(P )=2−ν
cβ (1 + 2ν |xP − xQ|)−M1p+β.
Applying Lemma 4.4 (Summation Lemma) to the sum on P , we have
Jν ≤ c
ν+1∑µ=ν−1
∑
l(Q)=2−µ
|Q|−p/2 ‖AQ~sQ‖p |Q| = c
ν+1∑µ=ν−1
∥∥∥∥∥∥∑
l(Q)=2−µ
|Q|−1/2 ~sQχQ
∥∥∥∥∥∥Lp({AQ},µ)
.
Combining the estimates for all Jν and reindexing when necessary, we get
∥∥∥~f∥∥∥
Bαqp ({AQ})
≤ 3 c
∥∥∥∥∥∥∥
2να
∥∥∥∥∥∥∑
l(Q)=2−ν
|Q|−1/2~sQχQ
∥∥∥∥∥∥Lp({AQ},ν)
ν
∥∥∥∥∥∥∥lq
= c ‖{~sQ}‖bαqp ({AQ}).
¥
Remark 7.8 Theorem 1.18 is obtained by combining Lemmas 7.5 and 7.7.
Corollary 7.9 If {AQ}Q is doubling (of order p), then
∥∥∥~f∥∥∥
Bαqp ({A−1
Q })≤ c
∥∥∥∥{~sQ(~f )
}Q
∥∥∥∥bαqp ({A−1
Q })(7.18)
and
∥∥∥~f∥∥∥
B−αq′p′ ({A−1
Q })≤ c
∥∥∥∥{~sQ(~f )
}Q
∥∥∥∥b−αq′p′ ({A−1
Q }). (7.19)
Proof. For (7.18) use the previous proof with the following shifting of AP to AQ
(similar to (7.14)):
‖A−1P ~sQ ‖p ≤ cn,β,p (1 + 2ν |xP − xQ|)β ‖A−1
Q ~sQ ‖p, (7.20)
86
where l(P ) = 2−ν and l(Q) = 2−µ with µ = ν − 1, ν or ν + 1; for (7.19) use the
above proof with the indices −α , q′ , p′ ; if 1 < p < ∞ , take M > βp′/p + n and
apply (7.20) raised to the power p′/p ; if p′ = ∞ , then
Jν ≤ supx∈Rn
ν+1∑µ=ν−1
∑
l(P )=2−ν
∑
l(Q)=2−µ
‖A−1P ~sQ‖ |(ϕν ∗ ψQ)(x)|χP (x).
Using the convolution estimate (7.17) (with M = M1 > β +n) and (7.20) for shifting
A−1P to A−1
Q , we get
Jν ≤ c
ν+1∑µ=ν−1
∥∥∥∥∥∥∑
l(Q)=2−µ
∥∥A−1Q ~sQ
∥∥ χQ
∥∥∥∥∥∥L∞
,
which gives (7.19). ¥
Corollary 7.10 The spaces Bαqp ({AQ}), Bαq
p ({A−1Q }) and B−αq′
p′ ({A−1Q }) are inde-
pendent of the choice of the admissible kernel, if {AQ}Q is doubling (of order p).
Proof. Repeat the proof of Theorem 1.8 with W replaced by AQ and use Lemmas
7.5 and 7.7 for the space Bαqp ({AQ}); for the space Bαq
p ({A−1Q }) apply (7.12) and
(7.18); and for the space B−αq′p′ ({A−1
Q }) use (7.13) and (7.19). ¥
7.4 Properties of averaging Lp spaces
In this section we study the connection between Lp({AQ}, ν) and Lp(W ) and the
dual of Lp({AQ}, ν).
Lemma 7.11 Let W be a doubling matrix weight of order p, 1 ≤ p < ∞. Then for
~f ∈ Eν , ν ∈ Z
‖~f ‖Lp(W ) ≤ c ‖~f ‖Lp({AQ},ν), (7.21)
where {AQ}Q is a sequence of reducing operators generated by W and c is independent
of ν .
87
Proof. Using the notation Wν(t) = W (2−νt) and ~fν(t) = ~f(2−νt), we write
‖~f ‖pLp(W ) =
∑
k∈Zn
∫
Qνk
‖W 1/p(t)~f(t)‖p dt =∑
k∈Zn
2−νn
∫
Q0k
‖W 1/pν (t)~fν(t)‖p dt.
Since ~fν ∈ E0 , there exists γ ∈ Γ such that ~fν = ~fν ∗ γ . Using the decay of γ and
Holder’s inequality, we get
‖~f ‖pLp(W ) ≤
∑
k∈Zn
2−νn
∫
Q0k
∑
m∈Zn
∫
Q0m
‖W 1/p(t)~fν(y)‖p
(1 + |m− k|)Mdy dt,
for some M > β + n . Observe that ‖AQνk~fν(y)‖p ≈ ∫
Q0k‖W 1/p
ν (t)~fν(y)‖p dt . Using
the doubling property of W to shift AQνkto AQνm (see (7.11)), we bound the previous
line by
c∑
m∈Zn
∑
k∈Zn
2−νn
∫
Q0m
(1 + |m− k|)−(M−β)‖AQνm~fν(y)‖p dy
≤ c∑
m∈Zn
∫
Q0m
‖AQνm~fν(y)‖p dy,
where the sum on k converges, since M > β + n . Changing variables x = 2−νy , we
get the desired inequality (7.21). ¥
Corollary 7.12 Let α ∈ R, 0 < q ≤ ∞ and 1 ≤ p < ∞. If W is doubling (of order
p) and {AQ}Q is a sequence of reducing operators generated by W , then
Bαqp ({AQ}) ⊆ Bαq
p (W ).
Proof. Since ϕν ∗ ~f ∈ Eν , the previous lemma implies
‖~f ‖Bαqp (W ) =
∥∥∥∥{
2να∥∥∥ϕν ∗ ~f
∥∥∥Lp(W )
}
ν
∥∥∥∥lq
≤ c
∥∥∥∥{
2να∥∥∥ϕν ∗ ~f
∥∥∥Lp({AQ},ν)
}
ν
∥∥∥∥lq
= c ‖~f ‖Bαqp ({AQ}).
¥
88
Lemma 7.13 Let 1 < p < ∞ and W satisfies any of (A1)-(A3). Suppose ~f ∈ Eν ,
ν ∈ Z. Then
‖~f ‖Lp({AQ},ν) ≤ c ‖~f ‖Lp(W ), (7.22)
where {AQ}Q is a sequence of reducing operators produced by W and c is independent
of ν .
Proof. Using the definition of reducing operators, we write
‖~f ‖Lp({AQ},ν) ≈∑
k∈Zn
∫
Qνk
1
|Qνk|∫
Qνk
‖W 1/p(t)~f(x) ‖p dt dx
=∑
k∈Zn
∫
Q0k
∫
Qνk
‖W 1/p(t)~fν(y) ‖p dt dy,
by changing variables x = 2−νy and denoting ~fν(y) = ~f(2−νy). Note that ~fν ∈ E0 .
Applying the decomposition of an exponential type function (Lemma 4.8) to ~fν =
~fν ∗ γ for γ ∈ Γ and Holder’s inequality (choose M > β + n), the last expression is
bounded by
c∑
k∈Zn
∫
Q0k
∫
Qνk
∑
m∈Zn
‖W 1/p(t)~fν(m)‖p
(1 + |y −m|)Mdt dy
≤ c∑
m∈Zn
∑
k∈Zn
1
(1 + |k −m|)M−β
∫
Q0k
∫
Qνm
‖W 1/p(t)~fν(m)‖p dt dy,
by applying the doubling property of W (any of (A1)-(A3) imply that W is doubling).
Integrating on y and summing on k (M > β + n), we obtain
‖~f‖pLp({AQ},ν) ≤ c
∑
m∈Zn
∫
Qνm
‖W 1/p(t)~fν(m)‖p dt
= c 2−νn∑
m∈Zn
∫
Q0m
‖W 1/pν (t)~fν(m)‖p dt,
again by changing variables. Now applying Lemma 4.12 and Lemma 4.14 (this
is where (A1)-(A3) come into play), we bound the above by c 2−νn ‖~fν ‖pLp(Wν) =
c ‖~f ‖pLp(W ) , which gives (7.22). ¥
89
Corollary 7.14 Let α ∈ R, 0 < q ≤ ∞ and 1 < p < ∞. If W satisfies any of
(A1)-(A3) and {AQ}Q is a sequence of reducing operators generated by W , then
Bαqp (W ) ⊆ Bαq
p ({AQ}).
Proof. As in the proof of Corollary 7.12, use the fact that ϕν ∗ ~f ∈ Eν and
Lemma 7.13. ¥
Remark 7.15 Combining Corollaries 7.12 and 7.14, we have Lemma 1.19.
In order to establish the dual of Lp({AQ}, ν), 1 < p < ∞ , we consider the following
idea:
‖~f ‖pLp({AQ},ν) =
∑Q∈Qν
∫
Q
‖AQ~f(x)‖p
H dx =
∫
Rn
( ∑Q∈Qν
‖AQ~f(x)‖HχQ(x)
)p
dx
=
∫
Rn
∥∥∥∥∥∑
Q∈Qν
AQχQ(x)~f(x)
∥∥∥∥∥
p
H
dx =:
∫
Rn
∥∥∥U1/pν (x)~f(x)
∥∥∥p
Hdx = ‖~f ‖p
Lp(Uν),
i.e., Lp({AQ}, ν) = Lp(Uν), where Uν(x) =∑
Q∈Qν
ApQχQ(x) is a matrix weight. Since
the dual [Lp(Uν)]∗ can be identified with Lp′(U∗
ν ) with U1/pν (x) = (U∗
ν )−1/p′(x) (e.g.
see [NT] or [V]), i.e., U∗ν (x) =
∑Q∈Qν
A−p′Q χQ(x), we obtain
‖~f ‖p′
Lp′ (U∗ν )=
∫
Rn
∥∥∥∥∥∑
Q∈Qν
A−1Q χQ(x)~f(x)
∥∥∥∥∥
p′
H
dx
=∑
Q∈Qν
∫
Q
‖A−1Q
~f(x)‖p′H dx = ‖~f ‖p′
Lp′ ({A−1Q },ν)
,
or
[Lp({AQ}, ν)]∗ ≈ Lp′({A−1Q }, ν). (7.23)
If p = 1, then the standard duality argument gives [L1({AQ}, ν)]∗ ≈ L∞({A−1Q }, ν).
The details are left to the reader.
90
7.5 Convolution estimates
The boundedness of the convolution operator with a decaying kernel on averaging
Lp({AQ}, ν) will be helpful in the next section. We establish it here.
Lemma 7.16 Let |Φ(t)| ≤ c
(1 + |t|)Mfor some M > β/p + n and for ν ∈ Z define
Φν(t) = 2νnΦ(2νt). Let {AQ}Q be a doubling matrix sequence of order p, 1 ≤ p < ∞.
Fix λ, µ, ν ∈ Z. Then
(i) ‖Φµ ∗ ~f ‖Lp({AQ},λ) ≤ c0 (c1)λ−ν(c2)
µ−ν‖~f ‖Lp({AQ},ν) ,
(ii) ‖Φµ ∗ ~f ‖Lp({A−1Q },λ) ≤ c0 (c3)
λ−ν(c2)µ−ν‖~f ‖Lp({A−1
Q },ν) ,
where c1 = 2n/pχ{λ>ν} + 2(n−β)/pχ{λ≤ν} , c2 = 2nχ{µ>ν} + 2n−Mχ{µ≤ν} , c3 =
2(β−n)/pχ{λ>ν} + 2−n/pχ{λ≤ν} , and c0 is independent of λ, µ and ν .
Proof. Using the decay of Φ, namely, |Φµ(x−y)| ≤ c k22νn
(1 + 2ν |x− y|)M, where
k2 = 2(µ−ν)nχ{µ>ν} + 2(ν−µ)(M−n)χ{µ≤ν} , we have
‖Φµ ∗ ~f ‖pLp({AQ},λ) =
∑Q∈Qλ
∫
Q
‖AQ(Φµ ∗ ~f )(x)‖p dx
≤∑
Q∈Qλ
∫
Q
(∫
Rn
‖AQ~f(y) ‖ |Φµ(x− y)| dy
)p
dx
≤ c∑
Q∈Qλ
∫
Q
(∫
Rn
k2 2νn‖AQ~f(y) ‖
(1 + 2ν |x− y|)Mdy
)p
dx
≈ c∑
k∈Zn
∫
Qλk
( ∑
m∈Zn
∫
Qνm
k2 2νn‖AQλk~f(y)‖
(1 + 2ν |x− xQνm|)Mdy
)p
dx.
Since {AQ}Q is doubling, we “shift” AQλkto AQνm :
‖AQλk~f(y) ‖ ≤ c k1(1+2ν |x−xQνm |)β/p ‖AQνm
~f(y) ‖, for x ∈ Qλk, (7.24)
91
where k1 = 2(λ−ν)n/pχ{λ>ν} + 2(ν−λ)(β−n)/pχ{λ≤ν} . Substituting (7.24) into the convo-
lution estimate, we get
‖Φµ ∗ ~f ‖pLp({AQ},λ) ≤ c
∫
Rn
( ∑
m∈Zn
∫
Qνm
k1k22νn‖AQνm
~f(y)‖(1 + |2νx−m|)M−β/p
dy
)p
dx.
Using the discrete Holder inequality on the sum inside and then Jensen’s inequality
to bring pth power inside of the integral (if p > 1), the last line is bounded above by
c kp1 kp
2
∫
Rn
(∑
l∈Zn
1
(1 + |2νx− l|)M−β/p
)p/p′ ( ∑
m∈Zn
∫
Qνm
2νn‖AQνm~f(y)‖p
(1 + |2νx−m|)M−β/pdy
)dx
≤ c kp1 kp
2
∑
m∈Zn
∫
Rn
2νn
(1 + |2νx−m|)M−β/p
∫
Qνm
‖AQνm~f(y)‖p dy dx,
since M−β/p > n , the sum on l converges (independently of x). Changing variables
(t = 2νx) and observing that the integral on t converges (independently of m), again
since M − β/p > n , we obtain
‖Φµ ∗ ~f ‖pLp({AQ},µ) ≤ c kp
1 kp2
∑
m∈Zn
∫
Qνm
‖AQνm~f(y)‖p dy.
Put c1 = k1/(λ−ν)1 and c2 = k
1/(µ−ν)2 . Then part (i) is proved.
For the second part observe that (1.7) (“shift” AQνm to AQλk) together with
‖A−1Q ~v ‖ = sup
~u6=0
|(~v, ~u )|‖AQ~u ‖ imply
‖A−1Qλk
~f(y) ‖ ≤ c k3(1 + 2ν |x− xQνm|)β/p‖A−1Qνm
~f(y) ‖, x ∈ Qλk, (7.25)
where k3 = 2(λ−ν)(β−n)/pχ{λ>ν}+2(ν−λ)n/pχ{λ≤ν} . Note that (7.25) is similar to (7.24),
so previous estimates with each AQ replaced by A−1Q prove (ii) with c3 = k
1/(λ−ν)3 .
¥
Remark 7.17 Recall that ‖A−1Q ~u ‖ ≤ c ‖A#
Q~u ‖ for any ~u ∈ H (since ‖(A#QAQ)−1‖ ≤
c). Suppose that W−p′/p is a doubling matrix of order p′ , 1 < p′ < ∞, with the
92
doubling exponent β∗ (instead of the assumption that W is doubling of order p).
Then
‖A−1Qλk
~f(y) ‖ ≤ c ‖A#Qλk
~f(y) ‖ ≤ c k∗1(1 + 2ν |x− xQνm|)β∗/p′‖A#Qνm
~f(y) ‖,
(where k∗1 = 2(λ−ν)n/p′χ{λ>ν} + 2(ν−λ)(β∗−n)/p′χ{λ≤ν} , i.e., k1 with β replaced by β∗
and p by p′ ) holds instead of (7.24). Choosing M > β∗/p′+n in the previous lemma,
we get
(iii) ‖Φµ ∗ ~f ‖Lp({A−1Q },λ) ≤ c0 (c∗1)
λ−ν(c2)µ−ν ‖~f ‖Lp({A#
Q},ν) , 1 < p < ∞.
Remark 7.18 A similar convolution estimate can be proved for Lp(W ) spaces, 1 <
p < ∞:
‖Φ ∗ ~f ‖Lp(W ) ≤ c ‖~f ‖Lp(W ). (7.26)
Recall that if Φ were to be a Calderon-Zygmund singular kernel K , then ‖K ∗~f ‖Lp(W ) ≤ c ‖~f ‖Lp(W ) if W ∈ Ap (see [NT], [TV1], [V]). Conversely, if (7.26)
holds for every Φ ∈ S , then W ∈ Ap is necessary (see the scalar case below).
Before we show the necessity of the Ap condition, we demonstrate that, for exam-
ple, having just a doubling weight is not sufficient for (7.26).
Example 7.19 Suppose w is a doubling scalar measure such that there exist E ⊂[0, 1] with 0 6= |E| = α < 1 but w(E) = 0. Such a measure exists (cf. [St2] or [FM]).
Let ϕ = χ[−1,1] . Choose f = χE . Then ‖f‖Lp(w) = 0.
Now (ϕ ∗ f)(x) =∫[x−1,x+1]
χE(t) dt = |[x − 1, x + 1] ∩ E|. So if x ∈ [0, 1], then
(ϕ ∗ f)(x) = |E| = α and if x /∈ [−1, 2], then (ϕ ∗ f)(x) = 0.
Thus, ‖ϕ∗f‖pLp(w) =
∫[−1,0]∪[1,2]
|(ϕ∗f)(x)|pw(x) dx+∫[0,1]�E
αpw(x) dx > 0, since
the second term is for sure positive.
93
Proposition 7.20 Let w be a scalar weight and 1 < p < ∞. Suppose that for every
Φ ∈ S the inequality
‖Φ ∗ f ‖Lp(w) ≤ CΦ ‖f ‖Lp(w) (7.27)
holds for any f ∈ Lp(w). Then
(1
|I|∫
I
w(y) dy
) (1
|I|∫
I
w−p′/p(y) dy
)p/p′
≤ c (7.28)
holds for any interval of side length l(I) ≤ cΦ , where cΦ is determined by Φ.
Remark 7.21 Observe that if (7.28) holds for any interval, then w ∈ Ap .
Proof of Proposition 7.20. Choose Φ ∈ S such that Φ(x) ≥ 1 for x ∈ [−2, 2].
Take f ≥ 0. Then (Φ ∗ f)(x) ≥∫
[x−2,x+2]
f(y) dy for x ∈ [−2, 2].
Consider I1, I2 ∈ D such that l(Ii) = 1, i = 1, 2, and I2 is right adjacent to
I1 . Let f = χI1 . Then for x ∈ I2 , we have (Φ ∗ f)(x) ≥∫
[x−2,x+2]
χI1(y) dy = 1,
and so ‖Φ ∗ f‖pLp(w) ≥
∫
I2
w(y) dy . Also ‖f‖pLp(w) =
∫
I1
w(y) dy . By (7.27) we get∫
I2
w(y) dy ≤ (CΦ) p
∫
I1
w(y) dy . By symmetry (suppose x ∈ I1 ), we get
∫
I1
w(y) dy ≤ (CΦ) p
∫
I2
w(y) dy. (7.29)
Note that the above two inequalities say that w is at least doubling.
Next, let f(y) = w−p′/p(y) χI1(y). Then for x ∈ I2 , we have (Φ ∗ f)(x) =∫
I1
w−p′/p(y) dy , and so ‖Φ ∗ f‖pLp(w) ≥
∫
I2
w(y) dy
(∫
I1
w−p′/p(y) dy
)p
. Also
‖f‖pLp(w) =
∫
I1
w−p′/p(y) dy . Again by (7.27),
∫
I2
w(y) dy
(∫
I1
w−p′/p(y) dy
)p
≤ (CΦ) p
∫
I1
w−p′/p(y) dy.
Substituting (7.29) and simplifying, we get
(∫
I1
w(y) dy
) (∫
I1
w−p′/p(y) dy
)p/p′
≤ (CΦ) 2p,
94
which is the scalar Ap condition for intervals of side length 1 (since I1 was arbitrary).
Now, let l(I1) = l(I2) = 2−ν , ν ≥ 0. Repeating the same argument as above
with f = χI1 and using symmetry, we get
∫
I1
w(y) dy ≤ (2ν CΦ) p
∫
I2
w(y) dy . Us-
ing f = w−p′/p χI1 again as before, we obtain
∫
I2
w(y) dy
(∫
I1
w−p′/p(y) dy
)p
≤
(2ν CΦ) p
∫
I1
w−p′/p(y) dy , and thus,
(1
|I1|∫
I1
w(y) dy
) (1
|I1|∫
I1
w−p′/p(y) dy
)p/p′
≤ (CΦ) 2p,
which gives (7.28) for intervals of side length ≤ 1 (cΦ = 1 in this case). ¥
7.6 Duality of continuous Besov spaces
Now we shift our attention to continuous Besov spaces and our task is to construct[Bαq
p ({AQ})]∗
and eventually[Bαq
p (W )]∗
.
Lemma 7.22 Let {AQ}Q be a doubling matrix sequence of order p, 1 ≤ p < ∞. Let
α ∈ R and 0 < q < ∞. Then
B−αq′p′ ({A−1
Q }) ⊆[Bαq
p ({AQ})]∗
. (7.30)
Proof. Take ϕ, ψ ∈ A with the mutual property (2.1). Let ψ(x) = ψ(−x).
Note that ˆψ(ξ) = ψ(ξ). Let ~f ∈ Bαqp ({AQ}) and ~g ∈ B−αq′
p′ ({A−1Q }). First consider
S0 = {f ∈ S : 0 /∈ supp f} a dense subspace of Bαqp ({AQ}) (see Appendix) and take
~f with (~f)i ∈ S0 , i = 1, ..., m (and ~g with (~g)i ∈ S ′ ). Then
~g =∑
ν∈Z~g ∗ (ϕν ∗ ψν), since
∑
ν∈Z(ϕν ∗ ψν)
ˆ(ξ) = 1, by (2.1),
and
~g (~f ) =∑
ν∈Z
(~g ∗ (ϕν ∗ ψν)
)(~f ) =
∑
ν∈Z(~g ∗ (ϕν ∗ ψν), ~f )
95
=∑
ν∈Z
((~g ∗ ϕν), (~f ∗ ψν)
)=
∑
ν∈Z
∫
Rn
⟨(~g ∗ ϕν)(x), (~f ∗ ψν)(x)
⟩H
dx
=∑
ν∈Z
∑Q∈Qν
∫
Q
⟨AQA−1
Q (~g ∗ ϕν)(x), (~f ∗ ψν)(x)⟩H
dx
≤∑
ν∈Z
∑Q∈Qν
∫
Rn
‖A−1Q (~g ∗ ϕν)(x)‖H ‖AQ(~f ∗ ψν)(x)‖HχQ(x) dx,
by the self-adjointness of each AQ and the Cauchy-Schwarz inequality. Using Holder’s
inequality several times, we obtain
|~g (~f )| ≤∑
ν∈Z2να
∥∥∥(~f ∗ ψν)∥∥∥
Lp({AQ},ν)· 2−να ‖(~g ∗ ϕν)‖Lp′ ({A−1
Q },ν) (7.31)
≤∥∥∥{
2να‖(~f ∗ ψν)‖Lp({AQ},ν)
}ν
∥∥∥lq
∥∥∥{
2−να‖(~g ∗ ϕν)‖Lp′ ({A−1Q },ν)
}ν
∥∥∥lq′ , if 1 < q < ∞,
and if 0 < q ≤ 1, we bound (7.31) by
∥∥∥{
2να‖(~f ∗ ψν)‖Lp({AQ},ν)
}ν
∥∥∥l1
∥∥∥{
2−να‖(~g ∗ ϕν)‖Lp′ ({A−1Q },ν)
}ν
∥∥∥l∞
≤∥∥∥{
2να‖(~f ∗ ψν)‖Lp({AQ},ν)
}ν
∥∥∥lq
∥∥∥{
2−να‖(~g ∗ ϕν)‖Lp′ ({A−1Q },ν)
}ν
∥∥∥l∞
.
Combining cases and using the independence of Bαqp ({AQ}) and B−αq′
p′ ({A−1Q }) on
the choice of the admissible kernel if {AQ}Q is doubling (Corollary 7.10), we get
|~g (~f )| ≤ ‖~f‖Bαqp ({AQ}) ‖~g ‖B−αq′
p′ ({A−1Q }).
Since S0 is dense in Bαqp ({AQ}), we get the above inequality for any ~f ∈ Bαq
p ({AQ}).Thus, ~g ∈ B−αq′
p′ ({A−1Q }) belongs to
[Bαq
p ({AQ})]∗
and ‖~g ‖oper ≤ ‖~g ‖B−αq′
p′ ({A−1Q }) .
¥
Lemma 7.23 Let α ∈ R, 1 ≤ p < ∞, 0 < q < ∞ and {AQ}Q be a doubling
sequence of order p Then
[Bαq
p ({AQ})]∗⊆ B−αq′
p′ ({A−1Q }). (7.32)
96
Proof. Let l ∈[Bαq
p ({AQ})]∗
. We show that there exists ~g ∈ B−αq′p′ ({A−1
Q })such that l(~f ) = ~g (~f ) = (~f,~g ) for any ~f ∈ Bαq
p ({AQ}).Case 1 ≤ q < ∞ . Take ~f ∈ Bαq
p ({AQ}), and for any ν ∈ Z denote ~fν = ~f ∗ ϕν .
Set T by T ({~fν}ν) = l(~f ), so T is defined on a subspace of lαq (Lp({AQ}, ν)) Since l
is bounded, so is T :
|T ({~fν}ν)| = |l(~f )| ≤ c ‖~f ‖Bαqp ({AQ}) = c ‖{~fν}ν‖lαq (Lp({AQ},ν)).
Extend T , denote the extension T , onto all of lαq (Lp({AQ}, ν)) (note: q ≥ 1).
Since [lq(X)]∗ ≈ lq′(X∗) (refer to Section 7.2 or [D, Ch.8]),
[lαq (Lp({AQ}, ν))
]∗
≈ l−αq′ ([Lp({AQ}, ν)]∗) ≈ l−α
q′ (Lp′({A−1Q }, ν)) by (7.23). Thus, there exists a vector-
valued sequence {~gν}ν∈Z ∈ l−αq′ (Lp′({A−1
Q }, ν)) such that ‖{~gν}ν∈Z‖l−αq′ (Lp′ ({A−1
Q },ν)) ≤‖l‖ and for any ~f ∈ Bαq
p ({AQ})
l(~f ) = T ({~fν}) = T ({~fν}) = {~gν}({~fν}) =∑
ν∈Z
∫
Rn
⟨~fν(x), ~gν(x)
⟩H
dx
=∑
ν∈Z
∫
Rn
〈(f ∗ ϕν)(x), ~gν(x)〉H dx =∑
ν∈Z
∫
Rn
⟨~f(x), (~gν ∗ ϕν)(x)
⟩H
dx.
Define ~g(x) =∑
ν∈Z(~gν ∗ ϕν)(x). Then l(~f ) = (~f,~g ). Moreover, for any ψ ∈ A (by
Corollary 7.10)
‖~g ‖B−αq′
p′ ({A−1Q }) ≈
∥∥∥∥∥∥∥
∥∥∥∥∥∑
ν∈Z~gν ∗ ϕν ∗ ψµ
∥∥∥∥∥Lp′ ({A−1
Q },µ)
µ
∥∥∥∥∥∥∥l−αq′
≤∥∥∥∥∥∥
{∑
ν∈Z‖~gν ∗ ϕν ∗ ψµ‖Lp′ ({A−1
Q },µ)
}
µ
∥∥∥∥∥∥l−αq′
=
∥∥∥∥∥∥
{µ+1∑
ν=µ−1
‖~gν ∗ ϕν ∗ ψµ‖Lp′ ({A−1Q },µ)
}
µ
∥∥∥∥∥∥l−αq′
,
97
since supp ψµ ⊆ {ξ : 2µ−1 ≤ |ξ| ≤ 2µ+1} and so ϕν∗ψµ = 0 if |µ−ν| > 1. Reindexing
the inner sum, we get
‖~g ‖q′
B−αq′p′ ({A−1
Q }) ≤ c∑
µ∈Z2−µαq′
1∑j=−1
‖~gµ+j ∗ ϕµ ∗ ψµ+j‖q′
Lp′ ({A−1Q },µ)
.
Since {AQ}Q is doubling and sum on j is finite, we apply Lemma 7.16 (ii) to get
‖~g ‖B−αq′
p′ ({A−1Q }) ≤ c′
∥∥∥∥{
2−µα‖~gµ‖Lp′ ({A−1Q },µ)
}µ
∥∥∥∥lq′≤ ‖l‖.
Case 0 < q < 1. Take ~f with (~f)i ∈ S0 . Since ϕ ∈ S0 , for ν ∈ Z by definition of
convolution and then boundedness of l , we have
|(l ∗ ϕν)(~f )| = |l(~f ∗ ϕν)| ≤ ‖l‖ ‖~f ∗ ϕν‖Bαqp ({AQ}). (7.33)
Note that each component of l ∗ ϕν is a C∞ -function and also ‖~f ∗ ϕν‖Bαqp ({AQ}) ≤
2να
ν+1∑µ=ν−1
‖~f ∗ ϕν‖Lp({AQ},µ) ≤ c 2να‖~f ‖Lp({AQ},ν) by Lemma 7.16 (i). Substituting this
estimate into (7.33), we get |(l ∗ ϕν)(~f )| ≤ c 2να‖l‖ ‖~f ‖Lp({AQ},ν) . By duality,
2−να‖l ∗ ϕν‖Lp′ ({A−1Q },ν) = 2−να sup
~f∈S0
|(l ∗ ϕν)(~f )|‖~f ‖Lp({AQ},ν)
≤ c ‖l‖,
i.e., the functional l ∗ ϕν can be associated with a function ~gν ∈ Lp′({A−1Q }, ν) such
that 2−να· ‖~gν‖Lp′ ({A−1Q },ν) ≤ c ‖l‖ . Let ~g =
∑ν∈Z ~gν ∗ θν , where θ is as in the atomic
decomposition theorem [FJW, Lemma 5.12], which implies ~g =∑
ν∈Z lϕν θν = l · 1and so g = l . Observe that ~g ∈ B−α∞
p′ ({A−1Q }):
‖g‖B−α∞p′ ({A−1
Q }) = supν∈Z
2−να‖g ∗ ϕν‖Lp′ ({A−1Q },ν) ≤ c ‖l‖.
Thus, the functional l ∈[Bαq
p ({AQ})]∗
can be associated with ~g ∈ B−α∞p′ ({A−1
Q })and l(~f ) = (~f,~g ). This completes the proof. ¥
Summarizing the results of the previous two sections we get the following embed-
dings of B -spaces:
98
Corollary 7.24 Let W be a matrix weight and {AQ}Q its reducing operators. Let
α ∈ R, 0 < q < ∞, 1 ≤ p < ∞. Then
[Bαq
p (W )]∗ (1)
⊆[Bαq
p ({AQ})]∗ (2)
⊆ B−αq′p′ ({A−1
Q })(3)
⊆ B−αq′p′ ({A#
Q})
(4)
⊆ B−αq′p′ (W−p′/p), (7.34)
where
(1) holds if W is doubling of order p, 1 < p < ∞,
(2) holds if W is doubling of order p, 1 ≤ p < ∞,
(3) holds if W ∈ Ap , 1 < p < ∞,
(4) holds if W−p′/p is doubling of order p′ , 1 < p < ∞.
Also,
[Bαq
p (W )]∗ (1∗)⊇
[Bαq
p ({AQ})]∗ (2∗)⊇ B−αq′
p′ ({A−1Q })
(3∗)⊇ B−αq′
p′ ({A#Q})
(4∗)⊇ B−αq′
p′ (W−p′/p), (7.35)
where
(1∗ ) holds if W satisfies any of (A1)-(A3),
(2∗ ) holds if W is doubling of order p, 1 ≤ p < ∞,
(3∗ ) holds for any matrix weight W ,
(4∗ ) holds if W−p′/p satisfies any of (A1)-(A3).
In terms of a matrix weight W only, (7.34) and (7.35) are
[Bαq
p (W )]∗⊆ B−αq′
p′ (W−p′/p) if W ∈ Ap, 1 < p < ∞
99
and
[Bαq
p (W )]∗⊇ B−αq′
p′ (W−p′/p) if W, W−p′/p satisfy any of (A1)-(A3).
In particular, if W ∈ Ap (and so W−p′/p ∈ Ap′ ), then[Bαq
p (W )]∗
≈B−αq′
p′ (W−p′/p), otherwise, (W still satisfies any of (A1)-(A3), or otherwise there
is a dependance on ϕ)[Bαq
p (W )]∗≈ B−αq′
p′ ({A−1Q }), which completes the proof of
Theorem 1.15.
7.7 Application of Duality
In this section we will study T ∗ψ and S∗ϕ and we will briefly show how duality can be
used to prove boundedness of operators Tψ and Sϕ . Recall that Tψ : bαqp (W ) −→
Bαqp (W ) by
s = {~sQ}Q 7−→∑Q
~sQ ψQ.
Moreover, Tψ is bounded if W is a doubling matrix of order p .
Let ~sQ ∈ bαqp (W ) and ~g ∈ Bαq
p (W ). Then
(~s, T ∗ψ~g) = (Tψ~s,~g) = (
∑Q
~sQ ψQ, ~g) =
∫ ⟨∑Q
~sQ ψQ(x), ~g(x)
⟩
H
dx
=∑Q
⟨~sQ,
∫ψQ(x)~g(x) dx
⟩
H=
∑Q
〈 ~sQ, (~g, ψQ)〉H =∑Q
〈 ~sQ, Sψ~g〉H = (~s, Sψ~g).
Therefore, T ∗ψ = Sψ and, similarly, S∗ϕ = Tϕ . So we have
T ∗ψ :
[Bαq
p (W )]∗−→
[bαqp (W )
]∗
or, another words,
Sψ : B−αq′p′ (W−p′/p) −→ b−αq′
p′ (W−p′/p),
and so Sψ is bounded if W ∈ Ap . Reformulating this by changing indices, we get
that under the Ap condition the following operators are bounded:
T ∗ψ :
[B−αq′
p′ (W−p′/p)]∗−→
[b−αq′p′ (W−p′/p)
]∗
100
or
Sψ : Bαqp (W ) −→ bαq
p (W ).
(This is another proof of Theorem 1.4.)
101
CHAPTER 8
Inhomogeneous Besov Spaces
8.1 Norm equivalence
In this section we discuss the inhomogeneous spaces. Before we define the vector-
valued inhomogeneous Besov space Bαqp (W ) with matrix weight W , we introduce a
class of functions A(I) with properties similar to those of an admissible kernel: we say
Φ ∈ A(I) if Φ ∈ S(Rn), supp Φ ⊆ {ξ ∈ Rn : |ξ| ≤ 2} and |Φ(ξ)| ≥ c > 0 if |ξ| ≤ 5
3.
Definition 8.1 (Inhomogeneous matrix-weighted Besov space Bαqp (W ))
For α ∈ R, 1 ≤ p < ∞, 0 < q ≤ ∞, W a matrix weight, ϕ ∈ A and Φ ∈ A(I) ,
we define the Besov space Bαqp (W ) as the collection of all vector-valued distributions
~f = (f1, ..., fm)T with fi ∈ S ′(Rn), 1 ≤ i ≤ m, such that
∥∥∥~f∥∥∥
Bαqp (W )
=∥∥∥Φ ∗ ~f
∥∥∥Lp(W )
+
∥∥∥∥∥{
2να∥∥∥ϕν ∗ ~f
∥∥∥Lp(W )
}
ν≥1
∥∥∥∥∥lq
< ∞.
Note that now we consider all vector-valued distributions in S ′(Rn) (rather than
S ′/P as in the homogeneous case), since Φ(0) 6= 0.
The corresponding inhomogeneous weighted sequence Besov space bαqp (W ) is de-
fined for the vector sequences enumerated by the dyadic cubes Q with l(Q) ≤ 1.
102
Definition 8.2 (Inhomogeneous weighted sequence Besov space bαqp (W ))
For α ∈ R, 1 ≤ p < ∞, 0 < q ≤ ∞ and W a matrix weight, the space bαqp (W )
consists of all vector-valued sequences ~s = {~sQ}l(Q)≤1 such that
‖~s ‖bαqp (W ) =
∥∥∥∥∥∥∥
2να
∥∥∥∥∥∥∑
l(Q)=2−ν
|Q|− 12~sQχQ
∥∥∥∥∥∥Lp(W )
ν≥1
∥∥∥∥∥∥∥lq
< ∞.
Following [FJ2], given ϕ ∈ A and Φ ∈ A(I) , we select ψ ∈ A and Ψ ∈ A(I) such
that
ˆΦ(ξ) · Ψ(ξ) +∑ν≥1
ˆϕ(2−νξ) · ψ(2−νξ) = 1 for all ξ, (8.1)
where Φ(x) = Φ(−x). Analogously to the ϕ-transform decomposition (2.2), we have
the identity for f ∈ S ′(Rn)
f =∑
l(Q)=1
〈f, ΦQ〉ΨQ +∞∑
ν=1
∑
l(Q)=2−ν
〈f, ϕQ〉ψQ, (8.2)
where ΦQ(x) = |Q|−1/2Φ(2νx− k) for Q = Qνk and ΨQ is defined similarly.
For each ~f with fi ∈ S ′(Rn) we define the inhomogeneous ϕ-transform S(I)ϕ :
Bαqp (W ) −→ bαq
p (W ) by setting (S(I)ϕ
~f )Q =⟨
~f, ϕQ
⟩if l(Q) < 1, and (S(I)
ϕ~f )Q =
⟨~f, ΦQ
⟩if l(Q) = 1.
The inverse inhomogeneous ϕ-transform T(I)ψ is the map taking a sequence
s = {sQ}l(Q)≤1 to T(I)ψ s =
∑
l(Q)=1
sQΨQ +∑
l(Q)<1
sQψQ . In the vector case, T(I)ψ ~s =
∑
l(Q)=1
~sQΨQ +∑
l(Q)<1
~sQψQ . By (8.2), T(I)ψ ◦ S
(I)ϕ is the identity on S ′(Rn).
Next we show that the relation between Bαqp (W ) and bαq
p (W ) is the same as for
the homogeneous spaces.
103
Theorem 8.3 Let α ∈ R, 0 < q ≤ ∞, 1 ≤ p < ∞, and let W satisfy any of
(A1)-(A3). Then
∥∥∥~f∥∥∥
Bαqp (W )
≈∥∥∥∥{~sQ
(~f)}
l(Q)≤1
∥∥∥∥bαqp (W )
. (8.3)
Before we outline the proof we need to adjust the notation of smooth molecules for
the inhomogeneous case. Define a family of smooth molecules {mQ}l(Q)≤1 for Bαqp (W )
as a collection of functions with the properties:
1. for dyadic Q with l(Q) < 1, each mQ is a smooth (δ,M,N)-molecule with
(M.i)-(M.iii) as for the homogeneous space Bαqp (W ) (see Section 4.1);
2. for dyadic Q with l(Q) = 1, each mQ (sometimes we denote it as MQ to
emphasize the difference) satisfies (M3), (M4) and a modification of (M2) (which
makes it a particular case of (M3) when γ = 0):
(M2∗ ) |mQ(x)| ≤ |Q|−1/2
(1 +
|x− xQ|l(Q)
)−M
.
Note that MQ does not necessarily have vanishing moments. Now one direction of
the norm equivalence (8.3) comes from the modified version of Theorem 4.2:
Theorem 8.4 Let α ∈ R, 1 ≤ p < ∞, 0 < q ≤ ∞ and W be a doubling matrix
weight of order p. Suppose {mQ}l(Q)≤1 is a family of smooth molecules for Bαqp (W ).
Then∥∥∥∥∥∥
∑
l(Q)≤1
~sQ mQ
∥∥∥∥∥∥Bαq
p (W )
≤ c ‖{~sQ}l(Q)≤1‖bαqp (W ). (8.4)
Sketch of the proof. We have∥∥∥∥∥∥
∑
l(Q)≤1
~sQ mQ
∥∥∥∥∥∥Bαq
p (W )
=
∥∥∥∥∥∥∑
l(Q)≤1
~sQ (Φ ∗mQ)
∥∥∥∥∥∥Lp(W )
104
+
∥∥∥∥∥∥∥
2να
∥∥∥∥∥∥∑
l(Q)≤1
~sQ (ϕν ∗mQ)
∥∥∥∥∥∥Lp(W )
ν≥1
∥∥∥∥∥∥∥lq
= I + II.
As in Theorem 4.2, which uses the convolution estimates (4.2) and (4.3), we need
similar inequalities for modified molecules (the proofs are routine applications of
Lemmas B.1 and B.2 from [FJ2]):
if l(Q) = 1, then
|Φ ∗MQ(x)| ≤ c (1 + |x− xQ|)−M , (8.5)
if l(Q) = 2−µ , µ ≥ 1, then for some σ > J − α
|Φ ∗mQ(x)| ≤ c |Q|− 12 2−µσ (1 + |x− xQ|)−M , (8.6)
if ν ≥ 1 and l(Q) = 1, then for some τ > α
|ϕν ∗MQ(x)| ≤ c 2−ντ (1 + |x− xQ|)−M , (8.7)
if ν ≥ 1 and l(Q) < 1, the estimate of |(ϕν ∗mQ)(x)| comes from either (4.2)
or (4.3).
To estimate I we use (8.5) and (8.6) (note that (8.5) is a special case of (8.6) for
µ = 0) and follow the steps of Theorem 1.10 by using Holder’s inequality twice to
bring the pth power inside of the sum, and the Squeeze and the Summation Lemmas
from Section 4.1 (it is essential that σ > J − α for convergence purposes) to get
I ≤ c ‖{~sQ}l(Q)≤1‖bαqp (W ).
The second term II is also estimated by ‖{~sQ}l(Q)≤1‖bαqp (W ) , which is obtained by
exact repetition of the proof of Theorem 4.2, only restricting the sum over µ ∈ Z to
105
the sum over µ ≥ 0. Also note that (8.7) is a particular case of (4.3) when µ = 0
and, thus, l(Q) = 1. Therefore, (8.4) is proved. ¥
In particular, since Φ and Ψ generate families of smooth molecules for Bαqp (W ),
we get∥∥∥~f
∥∥∥Bαq
p (W )≤ c
∥∥∥∥{~sQ
(~f)}
l(Q)≤1
∥∥∥∥bαqp (W )
,
which gives one direction of the norm equivalence (8.3). To show the other direction,
i.e., that the (inhomogeneous) ϕ-transform is bounded, we simply observe that Φ∗ ~f ∈E0 , which is true since
(Φ ∗ ~f
)∧i∈ S ′ and supp ˆΦ ⊆ {ξ ∈ Rn : |ξ| ≤ 2} . Hence,
Lemmas 4.12 and 4.14 apply to ~g = Φ ∗ ~f as stated. We have
∥∥∥∥{~sQ
(~f)}
l(Q)≤1
∥∥∥∥bαqp (W )
≈∥∥∥∥∥
∑
k∈Zn
(Φ ∗ ~f
)(k) χQ0k
∥∥∥∥∥Lp(W )
+
∥∥∥∥∥∥∥
2να
∥∥∥∥∥∥∑
l(Q)=2−ν
|Q|− 12 χQ
⟨~f, ϕQ
⟩∥∥∥∥∥∥
Lp(W )
ν≥1
∥∥∥∥∥∥∥lq
= I + II.
Using Φ ∗ ~f ∈ E0 and repeating the proof of Theorem 4.15 for both terms (in the
second term we take the lq norm only over ν ∈ N), we get the desired estimate:
∥∥∥∥{~sQ
(~f)}
l(Q)≤1
∥∥∥∥bαqp (W )
≤ c∥∥∥~f
∥∥∥Bαq
p (W ).
Note that as a consequence we also get independence of Bαqp (W ) from the choices of
Φ and ϕ .
8.2 Almost diagonality and Calderon-Zygmund
Operators
Now we will briefly discuss operators on the inhomogeneous spaces. An almost diag-
onal matrix on bαqp (W ) is the matrix A = (aQP )l(Q),l(P )≤1 whose entries satisfy (5.1),
106
i.e., |aQP | is bounded by (5.1) only for dyadic Q,P with l(Q), l(P ) ≤ 1. Such a
matrix A is a bounded operator on bαqp (W ) for the following reasons: let ~s ∈ bαq
p (W )
and then define ~s = {~sQ}Q∈D by setting ~sQ = ~sQ if l(Q) ≤ 1 and ~sQ = 0 if l(Q) > 1.
Note that ~s is a restriction of ~s on bαqp (W ). Also set A = (aQP )Q,P∈D putting
aQP = aQP if l(Q), l(P ) ≤ 1 and aQP = 0 otherwise. Then
‖A~s ‖bαqp (W ) =
∥∥∥∥∥∥∥
∑
l(P )≤1
aQP~sP
l(Q)≤1
∥∥∥∥∥∥∥bαqp (W )
=
∥∥∥∥∥∥
{ ∑
P dyadic
aQP~sP
}
Q
∥∥∥∥∥∥bαqp (W )
≤ c∥∥∥~s
∥∥∥bαqp (W )
,
by Theorem 1.10. By the construction, ‖~s ‖bαqp (W ) = ‖~s ‖bαq
p (W ) , and so we get bound-
edness of A on bαqp (W ).
It is easy to see that the class of almost diagonal matrices on bαqp (W ) is closed
under composition. The same statements (boundedness and being closed under com-
position) are true for the corresponding almost diagonal operators on Bαqp (W ) by
combining the norm equivalence (8.3) and the above results about almost diagonal
matrices on bαqp (W ). For Calderon-Zygmund operators on inhomogeneous matrix-
weighted Besov spaces, some minor notational changes should be made. The collec-
tion of smooth N -atoms {aQ}Q∈D in the homogeneous case ought to be replaced by
the set of atoms {aQ}l(Q)<1∪{AQ}l(Q)=1 , where the aQ ’s have the same properties as
before and the AQ ’s are such that supp AQ ⊆ 3Q and |DγAQ(x)| ≤ 1 for γ ∈ Zn+ .
This leads to a slight change of the smooth atomic decomposition (see [FJ2, p. 132]):
f =∑
l(Q)<1
sQ aQ +∑
l(Q)=1
sQ AQ.
With these adjustments, all corresponding statements about CZOs hold with essen-
tially the same formulations for the inhomogeneous spaces.
107
8.3 Duality
Let RS(I) be the collection of all sequences {AQ}l(Q)≤1 of positive-definite operators
on H . Similar to the homogeneous case, we introduce the averaging space bαqp ({AQ}).
Definition 8.5 (Inhomogeneous averaging matrix-weighted sequence
Besov space bαqp ({AQ}).) For α ∈ R, 0 < q ≤ ∞, 1 ≤ p ≤ ∞ and
{AQ}l(Q)≤1 ∈ RS(I) , let
bαqp ({AQ}) =
{~s =
{{~sQ}l(Q)≤1
}:
‖~s ‖bαqp ({AQ}) =
∥∥∥∥∥∥∥
2να
∥∥∥∥∥∥∑
l(Q)=2−ν
|Q|− 12~sQχQ
∥∥∥∥∥∥Lp({AQ},ν)
ν≥0
∥∥∥∥∥∥∥lq
< ∞
.
Let ~s ∈ bαqp (W ). Define ~s = {~sQ}Q∈D as in the previous section. Applying (1.10),
we get
‖~s ‖bαqp (W ) =
∥∥∥~s∥∥∥
bαqp (W )
≈∥∥∥~s
∥∥∥bαqp ({AQ})
= ‖~s ‖bαqp ({AQ}) ,
which proves the following proposition.
Proposition 8.6 Let α ∈ R, 1 ≤ p < ∞, 0 < q ≤ ∞ and let W be a matrix weight
with reducing operators {AQ}Q . Then
bαqp (W ) ≈ bαq
p ({AQ}),
in the sense of the norm equivalence.
Note that it’s enough to consider reducing operators AQ generated by a matrix weight
W only for dyadic cubes of side length l(Q) ≤ 1, i.e., {AQ}l(Q)≤1 .
Now we establish the duality.
108
Theorem 8.7 Let α ∈ R, 1 ≤ p < ∞, 0 < q < ∞ and let W be a matrix weight
with reducing operators {AQ}l(Q)≤1 . Then
[bαqp (W )
]∗ ≈ b−αq′p′ ({A−1
Q }).
Moreover, if W ∈ Ap , 1 < p < ∞, then
[bαqp (W )
]∗ ≈ b−αq′p′ (W−p′/p).
To prove this theorem, one can simply repeat the arguments from Section 7.2
with proper adjustments (for example, consider sums on ν taken only over ν ≥ 0).
However, we would like to give a simple proof for the embedding
[bαqp ({AQ})
]∗ ⊆ b−αq′p′ ({A−1
Q }).
Proof. Let l ∈ [bαqp ({AQ})
]∗. Let P be the projection from bαq
p ({AQ}) to bαqp ({AQ})
defined by restricting a sequence {~sQ}Q∈D to {~sQ}l(Q)≤1 . Set l by l(~s ) = l(P~s ) for
each ~s ∈ bαqp ({AQ}). Then l ∈
[bαqp ({AQ})
]∗, since
|l(~s )| = |l(P~s )| ≤ ‖l‖ ‖P~s ‖bαqp ({AQ}) ≤ ‖l‖ ‖~s ‖bαq
p ({AQ}).
Then by Lemma 7.3 (or, equivalently, by (7.6)), l is represented by ~t ∈ b−αq′p′ ({A−1
Q })such that l(~s ) = (~s, ~t ) and ‖~t ‖
b−αq′p′ ({A−1
Q }) ≤ ‖l‖ ≤ ‖l‖ . Let ~t = P~t . For ~s ∈bαqp ({AQ}) define ~s ∈ bαq
p ({AQ}) as above (i.e., P~s = ~s ). Then
l(~s ) = l(~s ) = (~s, ~t ) =∑
l(Q)≤1
~sQ~tQ +
∑
l(Q)>1
~sQ~tQ =
∑
l(Q)≤1
~sQ~tQ = (~s,~t ),
since ~sQ = 0 for l(Q) > 1. Moreover, ‖~t ‖b−αq′p′ ({A−1
Q }) ≤ ‖~t ‖b−αq′p′ ({A−1
Q }) ≤ ‖l‖ . ¥
Analogously, we introduce the averaging space Bαqp ({AQ}).
Definition 8.8 (Averaging matrix-weighted Besov space Bαqp ({AQ})) For
α ∈ R, 0 < q ≤ ∞, 1 ≤ p ≤ ∞, ϕ ∈ A, Φ ∈ A(I) and {AQ}l(Q)≤1 ∈ RS(I) , let
Bαqp ({AQ}) =
{~f = (f1, ..., fm)T with fi ∈ S ′(Rn), 1 ≤ i ≤ m :
109
∥∥∥~f∥∥∥
Bαqp ({AQ})
=∥∥∥Φ ∗ ~f
∥∥∥Lp({AQ},0)
+
∥∥∥∥∥{
2να∥∥∥ϕν ∗ ~f
∥∥∥Lp({AQ},ν)
}
ν≥1
∥∥∥∥∥lq
< ∞}
.
Now the remaining results from Sections 7.3, 7.4 and 7.6 transfer easily to the
inhomogeneous Besov spaces by using the properties (discussed earlier in this section)
such as replacing a family {ϕν}ν∈Z with {ϕν}ν∈N∪Φ; observing that Φ ∗ ~f ∈ E0 and
summing over ν ≥ 0 (or l(Q) ≤ 1) in all sums. In particular, we get
Theorem 8.9 Let α ∈ R, 0 < q ≤ ∞, 1 ≤ p < ∞ and let {AQ}l(Q)≤1 be a doubling
sequence (of order p). Then for ~sQ(~f ) =⟨
~f, ϕQ
⟩
∥∥∥~f∥∥∥
Bαqp ({AQ})
≈∥∥∥∥{~sQ
(~f)}
l(Q)≤1
∥∥∥∥bαqp ({AQ})
.
Corollary 8.10 The spaces Bαqp ({AQ}), Bαq
p ({A−1Q }) and B−αq′
p′ ({A−1Q }) are inde-
pendent of the choice of the pair of admissible kernels (ϕ, Φ), if {AQ}l(Q)≤1 is doubling
(of order p), 1 ≤ p < ∞, α ∈ R, 0 < q ≤ ∞.
Lemma 8.11 Let α ∈ R, 0 < q ≤ ∞ and 1 ≤ p < ∞. If W satisfies any of
(A1)-(A3) and {AQ}l(Q)≤1 is a sequence of reducing operators generated by W , then
Bαqp (W ) ≈ Bαq
p ({AQ}).
Theorem 8.12 Let α ∈ R, 0 < q < ∞, 1 ≤ p < ∞ and let {AQ}l(Q)≤1 be reducing
operators of a matrix weight W . If W ∈ Ap , 1 < p < ∞, then
[Bαq
p (W )]∗ ≈ B−αq′
p′ (W−p′/p).
If W satisfies any of (A1)-(A3), then
[Bαq
p (W )]∗ ≈ B−αq′
p′ ({A−1Q }).
Thus, all results obtained for the matrix-weighted homogeneous Besov spaces are
essentially the same for the inhomogeneous case.
110
CHAPTER 9
Weighted Triebel-Lizorkin Spaces
9.1 Motivation
The study in this chapter was stimulated by the question posed by A. Volberg in [V].
He proved that if W ∈ Ap , 1 < p < ∞ , then the following equivalence takes place
∥∥∥∥{
sQ
(~f)}
Q
∥∥∥∥f02
p (W )
≈∥∥∥∥{
sQ
(~f)}
Q
∥∥∥∥f02
p ({AQ}), (9.1)
where sQ(~f ) =⟨
~f, hQ
⟩with {hQ}Q being the well-known Haar system and {AQ}Q
the reducing operators for W . Moreover, he pointed out that the equivalence does not
necessarily require W ∈ Ap . For example, it holds always for p = 2. He conjectured
that for p ≥ 2, the condition on the metric ρ generated by W , which is similar to a
scalar A∞ condition, ρ ∈ Ap,∞ might be sufficient. The criterion for (9.1) was asked.
In the light of our studies of function spaces, we rephrase (and partially answer)
the question of Volberg in familiar terms: what conditions on W are needed for the
equivalence below to hold?
∥∥∥{~sQ}Q
∥∥∥fαq
p (W )≈
∥∥∥{~sQ}Q
∥∥∥fαq
p ({AQ})(9.2)
Our main result of this chapter deals with scalar weights and the matrix case is
left for future research. We show in Theorem 9.3 that if a scalar weight w ∈ A∞ , then
111
(9.2) holds for α ∈ R , 0 < q ≤ ∞ and 0 < p < ∞ . Furthermore, by using the result of
Frazier and Jawerth (see [FJ2, Proposition 10.14]), we connect the reducing operators
sequence space fαqp ({wQ}) with the continuous Triebel-Lizorkin space Fαq
p (w), and
therefore, obtain the following norm equivalence
‖f‖F αqp (w) ≈ ‖{〈f, ϕQ〉}Q‖fαq
p ({wQ}).
9.2 Equivalence of fαqp (w) and fαq
p ({wQ})
Before we prove the main result, we establish two lemmas for the weighted and
unweighted maximal functions. Denote wQ = 1|Q|
∫Q
w(x) dx .
Lemma 9.1 Let EQ = {x ∈ Q : w(x) ≤ 2wQ}. If w ∈ A∞ , then Mw(χEQ) ≥ c χQ .
Proof. Using the definition of the maximal function, for x ∈ Q we have
Mw(χEQ)(x) = sup
I:x∈I
1
w(I)
∫
I
χEQ(y)w(y) dy ≥ 1
w(Q)
∫
Q
χEQ(y)w(y) dy =
w(EQ)
w(Q).
The condition w ∈ A∞ implies thatw(EQ)
w(Q)≥ c
( |EQ||Q|
)β
for some β > 0. Since
EQ = {x ∈ Q : w(x) ≤ 2wQ} , the compliment EcQ = {x ∈ Q : w(x) > 2wQ} . This
gives us the following chain of inequalities:
w(Q) =
∫
Q
w(x) dx ≥∫
EcQ
w(x) dx >
∫
EcQ
2wQ dx = 2wQ|EcQ|.
In other words, |Q|wQ = w(Q) > 2wQ|EcQ| , or |Q| > 2|Ec
Q| , which implies |EQ| >
12|Q| . Hence, for x ∈ Q we have
Mw(χEQ)(x) ≥ c
(1
2
)β
= c′ χQ(x),
which finishes the proof. ¥
112
Lemma 9.2 There exists 0 < δ < 1 such that if EQ = {x ∈ Q : w(x) ≥ δ wQ} and
w ∈ A∞ , then M(χEQ) ≥ c χQ .
Proof. The proof goes similarly to the proof of the first lemma except we will
apply the A∞ condition in a slightly different way. If x ∈ Q , then
M(χEQ)(x) = sup
I:x∈I
1
|I|∫
I
χEQ(y) dy ≥ 1
|Q|∫
Q
χEQ(y) dy ≥ |EQ|
|Q| .
Considering the compliment of EQ , we have EcQ = {x ∈ Q : w(x) < δ wQ} . Then,
w(EcQ) =
∫
EcQ
w(x) dx <
∫
EcQ
δ wQ dx ≤ δ wQ
∫
Q
dx = δ wQ|Q| = δ w(Q).
So w(EcQ) ≤ δ w(Q) implies w(EQ) ≥ (1− δ) w(Q). Since w ∈ A∞ and
w(EcQ)
w(Q)≤ δ ,
there exists ε < 1 such that|Ec
Q||Q| ≤ ε or
|EQ||Q| ≥ 1 − ε . Hence, for x ∈ Q we have
M(χEQ)(x) ≥ (1− ε), or
M(χEQ)(x) ≥ (1− ε)χQ(x).
¥
Theorem 9.3 Suppose w ∈ A∞ and let α ∈ R, 0 < q ≤ ∞ and 0 < p < ∞. Then
‖{sQ}Q‖fαqp ({wQ}) ≈ ‖{sQ}Q‖fαq
p (w).
Moreover, if f ∈ Fαqp (w), then for sQ(f) = 〈f, ϕQ〉,
‖f‖F αqp (w) ≈ ‖{sQ(f)}Q‖fαq
p ({wQ}). (9.3)
Proof. By definition,
‖{sQ(f)}Q‖fαqp (w) =
∥∥∥∥∥∥
(∑Q
(|Q|− 12−α
n |sQ|wχQ)q
)1/q∥∥∥∥∥∥
Lp
.
113
We need to show that the last norm is equivalent to
∥∥∥∥{
sQ · w1/pQ
}Q
∥∥∥∥fαq
p
=
∥∥∥∥∥∥
(∑Q
(|Q|− 12−α
n |sQ|w1/pQ χQ)q
)1/q∥∥∥∥∥∥
Lp
.
Let EQ = {x ∈ Q : w(x) ≤ 2wQ} . Choose A > 0 such that p/A > 1 and q/A > 1.
By Lemma 9.1, χQ(x) ≤ c(Mw(χA
EQ)(x)
)1/A
. Therefore,
‖{sQ}Q‖fαqp (w) =
∫ [∑Q
(|Q|− 12−α
n |sQ|χQ(x))q
]p/q
w(x) dx
1/p
≤ c
∫ [∑Q
(|Q|− 1
2−α
n |sQ|(Mw(χA
EQ)(x)
)1/A)q
]p/q
w(x) dx
1/p
≤ c
∫ [∑Q
(Mw(|Q|− 1
2−α
n |sQ|χEQ(x))A
)q/A]p/q
w(x) dx
1/p
≤ c
∥∥∥∥∥∥
[∑Q
(Mw(|Q|− 1
2−α
n |sQ|χEQ)A(x)
)q/A]A/q
∥∥∥∥∥∥
1/A
Lp/A(w)
.
Since w is a doubling measure and the weighted maximal function Mw satisfies the
vector-valued maximal inequality (see [St1] or [St2]), the last expression is bounded
above by
cp,q
∥∥∥∥∥∥
[∑Q
(|Q|− 12−α
n |sQ|χEQ)q
]A/q∥∥∥∥∥∥
1/A
Lp/A(w)
= cp,q
∫ [∑Q
(|Q|− 12−α
n |sQ|χEQ(x)w1/p(x))q
]p/q
dx
1/p
≤ 21/pcp,q
∫ [∑Q
(|Q|− 12−α
n |sQ|w1/pQ χEQ
(x))q
]p/q
dx
1/p
≤ c
∫ [∑Q
(|Q|− 12−α
n |sQ|w1/pQ χQ(x))q
]p/q
dx
1/p
= c
∥∥∥∥{
sQw1/pQ
}Q
∥∥∥∥fαq
p
.
In the last inequality we used EQ ⊆ Q .
114
For the opposite direction, set EQ = {x ∈ Q : w(x) ≥ δwQ} and again choose
A > 0 such that p/A > 1 and q/A > 1. Then by Lemma 9.2,
∫ [∑Q
(|Q|− 12−α
n |sQ|w1/pQ χQ(x))q
]p/q
dx
1/p
≤ c
∫ [∑Q
(|Q|− 12−α
n |sQ|w1/pQ M(χA
EQ)(x))q
]p/q
dx
1/p
= c
∫ [∑Q
(M(|Q|− 1
2−α
n |sQ|w1/pQ χEQ
(x))A)q/A
]p/q
dx
1/p
≤ δ1/pc
∫ [∑Q
(|Q|− 12−α
n |sQ|w1/p(x)χEQ(x))q
]p/q
dx
1/p
≤ c
∫ [∑Q
(|Q|− 12−α
n |sQ|χQ(x))q
]p/q
w(x) dx
1/p
= c ‖{sQ}Q‖fαqp (w) .
It is easy to show the second assertion. By [FJ1, Proposition 10.14]
‖f‖F αqp (w) ≈ ‖{sQ(f)}Q‖fαq
p (w).
Combining this equivalence with the first result, we get (9.3). The proof is complete.
¥
115
CHAPTER 10
Open Questions
1. In the unweighted theory of function spaces, special cases of the Besov and
Triebel-Lizorkin spaces are the Lebesgue spaces: e.g. B022 = L2 and F 02
p = Lp ,
1 < p < +∞ . In the scalar weighted situation it is known that B022 (w) = L2(w)
and F 02p (w) = Lp(w) if and only if w ∈ Ap . In the matrix case it is expected that
the Ap condition is the minimal condition on W needed for this equivalence
to hold. Using vector-valued square function operators might be one of the
approaches to this problem.
2. The crucial step for our theory of Besov spaces is the norm equivalence between
continuous Bαqp (W ) and discrete bαq
p (W ) spaces. We were able to show that it
holds for any doubling matrix weights if the order p is greater than the doubling
exponent of W . In the special case of diagonal matrices (equivalently, in the
scalar case) this restriction is removed. A conjecture is that the equivalence
holds for any doubling matrix weight W .
3. One major goal is to answer the same “equivalence of norms” questions for
the matrix-weighted Triebel-Lizorkin spaces Fαqp (W ) and fαq
p (W ). The scalar
weighted case is known (see [FJ2], [FJW]) and the matrix-weighted case is to
116
be studied. This will also lead to the question of the boundedness of singular
integral operators on Fαqp (W ). Possible approaches include a variation of the
exponential type estimates used in the Besov space case or Volberg’s factoriza-
tion method mentioned in the introduction.
4. All previous research was done on function spaces with the parameter p being
between 1 and ∞ , quite often not including the end point p = 1, which requires
more careful consideration. Moreover, this raises the question of matrix “A1”
weights and their factorization, and whether this can be developed further into
an extrapolation theory of matrix-weighted distribution spaces (similar to Rubio
de Francia results in scalar case). Furthermore, it would be interesting to study
a case when 0 < p < 1. Nothing is known in this area, except for certain scalar
cases.
5. The motivation for the norm equivalence studied above came from the fact
that ‖~f ‖Lp(W ) ≈ ‖{< ~f, hI >}I‖f02p (W ) , where {hI}I is a Haar system and
{< ~f, hI >}I constitutes a sequence of the Haar coefficients of ~f . Recall that
we obtained the norm equivalence when the generators of the expansion (either
the ϕ-transform or wavelet functions) have some degree of smoothness. This
property is lacking for the Haar system. Nevertheless, the Haar system is widely
used in applications. This creates the open question (for ceirtain indices α , q ,
p) of the norm equivalence between continuous and discrete function spaces
with the Haar coefficients.
6. A very difficult problem of modern Fourier analysis is to obtain weighted norm
inequalities on the function spaces (at least on Lp spaces) with different weights.
Complete answers to the scalar two-weight problem is known only for the Hardy-
117
Littlewood maximal function (by Muckenhoupt and Wheeden in [MW] and by
Sawyer in [Sa]). For the Hilbert Transform the necessary condition is given by
Muckenhoupt and Wheeden and the dyadic version is studied by Nazarov, Treil
and Volberg in [NTV]. Furthermore, the necessary and sufficient conditions for
the case p = 2 are obtained by Cotlar and Sadosky in [CS]. Since the Lebesgue
spaces are special cases of Besov and Triebel-Lizorkin function spaces, the same
two-weight questions should be asked in the light of Littlewood-Paley theory.
The hope is to consider at least the scalar case and to obtain the conditions
on the two weights for the boundedness of the ϕ-transform, almost diagonal
operators, maximal function operators (such as Peetre’s maximal operator), the
Hilbert Transform and possibly other singular integral operators.
118
Appendix A
Density and convergence
Define S0 = {f ∈ S : 0 /∈ supp f} . Observe that ψ ∈ A implies ψ, ψν , ψQ ∈ S0 for
ν ∈ Z and Q dyadic.
Lemma A.1 Let f ∈ S0 . Then fN :=∑
|ν|≤N
ϕν ∗ ψν ∗ f =∑
|ν|≤N
〈f, ϕQνk〉 ψQνk
con-
verges to f in the S -topology as N →∞.
Proof. For ν ∈ Z , define f(ν) = ϕν ∗ ψν ∗ f =∑
Q∈Qν
〈f, ϕQ〉ψQ . Then f(ν)(ξ) =
ˆϕν(ξ)ψν(ξ)f(ξ) and ϕν , ψν , f ∈ S0 =⇒ f(ν) ∈ S0 . Observe another fact: since f ∈ S0 ,
there exists N0 ∈ N large such that f =∞∑
ν=−N0
f(ν) . Indeed, 0 /∈ supp f implies that
f(x) = 0 if |x| ≤ 2−N0 for some N0 > 0. Thus, for large N , fN =N∑
ν=−N0
f(ν) =
∑ν≤N
f(ν) .
To prove the lemma, it suffices to show that ργ(fN − f) → 0 as N →∞ . Denote
mN(ξ) = 1−N∑
ν=−N0
ˆϕν(ξ)ψν(ξ). Because of the support of ϕν and ψν and the mutual
property (2.1), mN(ξ) = 0 for 2−N0 < |ξ| < 2N . Moreover, 0 ≤ mN(ξ) ≤ 1 for
2N < |ξ| < 2N+1 and mN(ξ) = 1 for |ξ| ≥ 2N+1 . Using these facts, we obtain
ργ(fN − f) = sup|ξ|≥2n, |α|≤γ
(1 + |ξ|2)γ|Dα(mN f)(ξ)|
119
≤ c sup|ξ|≥2N , |α|≤γ
(1 + |ξ|2)γ∑
γ1+γ2=α
|Dγ1mN(ξ)| |Dγ2 f(ξ)|.
Observe that |Dγ1mN(ξ)| ∼ 1/|ξ| |γ1| if |ξ| ∼ 2N and that f ∈ S implies |Dγ2 f(ξ)| ≤cL
(1 + |ξ|)L+|γ2| for any L > 0. Take L > 2γ . Then
ργ(fN − f) ≤ cL sup|ξ|≥2N , |α|≤γ
∑γ1+γ2=α
(1 + |ξ|)2γ
(1 + |ξ|)|γ1| (1 + |ξ|)L+|γ2| ≤cL
(1 + 2N)L−2γ−→N→∞
0.
Thus, fN −→N→∞
f in S -topology. ¥
Lemma A.2 Let f ∈ S0 and fix ν ∈ Z. Then∑
|k|≤M
〈f, ϕQνk〉 ψQνk
converges to
fν = ϕν ∗ ψν ∗ f =∑
k∈Zn
〈f, ϕQνk〉 ψQνk
in S -topology as M →∞.
Proof. Denote fν,M =∑
k≤M
〈f, ϕQνk〉ψQνk
. Obviously, fν,M ∈ S0 . Then
ργ(fν,M − fν) = supx∈Rn, |α|≤γ
(1 + |x|2)γ |Dα(fν,M − fν)(x)|
= supx∈Rn, |α|≤γ
(1 + |x|2)γ
∣∣∣∣∣Dα
∑
k>M
〈f, ϕQ〉ψQ(x)
∣∣∣∣∣ .
Observe that we can bring Dα inside of the sum. (A similar argument as below proves
this claim.) Then
ργ(fν,M − fν) ≤ supx∈Rn, |α|≤γ
(1 + |x|)2γ∑
|k|>M
|ϕν ∗ f(2−νk)| |Dα(ψ(2νx− k))|
(A.1)
Choose L1 > 2γ − |α| , L2 > 2γ + n and L3 > max(L1, γ). Using properties of Sfunctions, we have
|Dα (ψ(2νx− k))| = 2ν|α| |(Dαψ) (2νx− k)| ≤ cL1,α 2ν|α|
(1 + |2νx− k|)L1+|α| .
Applying the convolution estimates (4.2) and (4.3), we bound (ϕν ∗ f)(k):
|(ϕν ∗ f)(k)| ≤ cL2 2−|ν|L3
(1 + |k|)L2.
120
Substituting the above estimates into (A.1), we obtain
ργ(fν,M − fν) ≤ supx∈Rn,|α|≤γ
∑
|k|>M
cL1,L2,α (1 + |x|)2γ 2ν|α|−|ν|L3
(1 + |k|)L2 (1 + |2νx− k|)L1+|α|
≤ cL1,L2,α
(sup
x∈Rn,|α|≤γ
(1 + |x|)2γ 2ν|α|−|ν|L3
(1 + 2ν |x|)L1+|α|
) ∑
|k|>M
1
(1 + |k|)L2−2γ,
by using (1+2ν |x|) ≤ (1+ |2νx−k|) (1+ |k|). The supremum on x and α is bounded
by cν = 2ν(γ−L3)χ{ν≥0} + 2ν(L3−L1)χ{ν<0} , since L1 > 2γ − |α| . Thus, we get
ργ(fν,M − fν) ≤ cL1,L2,γ cν
∑
|k|>M
1
(1 + |k|)L2−2γ−→ 0 as M −→∞
as a tail of a convergent series, since L2−2γ > n . Thus, fν,M −→M→∞
fν in S -topology.
¥
Remark A.3 If T is a continuous linear operator from S into S ′ and f ∈ S0 , then
Tf =∑
ν∈ZT (ϕν ∗ ψν ∗ f) =
∑
ν∈Z
∑
k∈Zn
〈f, ϕQνk〉TψQνk
=∑Q
〈f, ϕQ〉TψQ.
Lemma A.4 S0 is dense in Bαqp (W ) for α ∈ R, 0 < q < ∞, 1 ≤ p < ∞ and if W
satisfies any of (A1)-(A3).
Proof. Let ~f ∈ Bαqp (W ). For N ∈ N denote ~f N =
N∑ν=−N
∑Q∈Qν
⟨~f, ϕQ
⟩ψQ . Then
by Corollary 4.6
∥∥∥~f − ~f N∥∥∥
q
Bαqp (W )
=
∥∥∥∥∥∥∑
|ν|>N
∑Q∈Qν
~sQ
(~f)∥∥∥∥∥∥
q
Bαqp (W )
≤ c
∥∥∥∥{~sQ
(~f)}
Q∈Qν ,|ν|>N
∥∥∥∥q
bαqp (W )
= c∑
|ν|>N
2να
∥∥∥∥∥∑
Q∈Qν
|Q|−1/2~sQχQ
∥∥∥∥∥Lp(W )
q
−→N→∞
0
as a tail of the convergent series
∑
ν∈ZAν :=
∑
ν∈Z
2να
∥∥∥∥∥∑
Q∈Qν
|Q|−1/2~sQχQ
∥∥∥∥∥Lp(W )
q
= ‖{~sQ}Q‖q
bαqp (W )
(A.2)
121
≤ c ‖~f ‖q
Bαqp (W )
< ∞,
if W satisfies any of (A1)-(A3) by Theorem 4.15.
As in the previous proposition for each ν ∈ Z and M ∈ N define ~fν,M =∑
k≤M
⟨~f, ϕQνk
⟩ψQνk
∈ S0 and recall ~fν =∑
Q∈Qν
⟨~f, ϕQ
⟩ψQ . Note that ~f N =
∑
|ν|≤N
~fν .
Then
∥∥∥~fν − ~fν,M
∥∥∥Bαq
p (W )≤ c
∥∥∥∥{~sQνk
(~f)}
|k|≥M
∥∥∥∥bαqp (W )
= c 2να
∥∥∥∥∥∥∑
|k|≥M
2νn/2~sQνkχQνk
∥∥∥∥∥∥Lp(W )
= c 2να
∑
|k|≥M
2νnp/2
∫
Qνk
‖W 1/p(t)~sQνk‖p dt
1/p
−→M→∞
0
again as a tail of the convergent series∑
|k|∈Zn
2νnp/2
∫
Qνk
‖W 1/p(t)~sQνk‖p dt =
(2−ναA1/qν )p < ∞ , since each Aν < ∞ (see (A.2)). Thus, each ~f ∈ Bαq
p (W ) is a
limit (in Bαqp (W )-norm) of S0 functions and so S0 is a dense subset of Bαq
p (W ).
¥
Proposition A.5 S0 is dense in Bαqp ({AQ}) if {AQ}Q is doubling of order p, 1 ≤
p < ∞ and α ∈ R, 0 < q < ∞.
Proof. Repeat the previous proof with W replaced by {AQ}Q and refer to
Lemma 7.7 instead of Corollary 4.6 and to Lemma 7.5 instead of Theorem 4.15. Both
require {AQ}Q to be only doubling. ¥
122
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123
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