real interpolation of generalized besov-hardy spaces and applications

J Fourier Anal Appl (2011) 17:691–719DOI 10.1007/s00041-010-9145-2

Real Interpolation of Generalized Besov-Hardy Spacesand Applications

Alexandre Almeida · António Caetano

Received: 11 February 2010 / Published online: 22 September 2010© Springer Science+Business Media, LLC 2010

Abstract In this paper we consider generalized Hardy spaces which include classicalHardy spaces and Hardy-Lorentz spaces as special cases. We give real interpolationresults for such spaces. As applications, we solve an interpolation problem for Besovspaces of generalized smoothness and prove the boundedness of pseudodifferentialoperators acting both in these spaces and in the local Hardy spaces. For the latterspaces, we also obtain wavelet decompositions.

Keywords Generalized Hardy spaces · Maximal functions · Interpolation · Besovspaces · Pseudodifferential operators · Wavelet decompositions

Mathematics Subject Classification (2000) 46E35 · 42C40 · 42B30 · 46M35 ·35S05

1 Introduction

The real maximal theory of Hardy spaces Hp , 0 < p < ∞, has started in the begin-ning of the seventies of last century, mainly with the work of Fefferman and Stein[16]. It is known that the Hp spaces have better functional properties than the Lp

spaces when 0 < p < 1. For instance, Hp spaces are stable under the action of im-portant classes of singular operators. Nevertheless, these spaces still miss some im-portant properties. For example, Hp does not behave well under multiplication by

Communicated by Hans Triebel.

Research partially supported by the Research Unit Matemática e Aplicações (UIMA) of Universityof Aveiro.

A. Almeida (�) · A. CaetanoDepartamento de Matemática, Universidade de Aveiro, 3810-193 Aveiro, Portugale-mail: [email protected]

A. Caetanoe-mail: [email protected]

mailto:[email protected]

mailto:[email protected]

692 J Fourier Anal Appl (2011) 17:691–719

test functions. In order to overcome this and other difficulties, Goldberg [18] intro-duced the hp spaces, a local version of Hardy spaces. Notably, important classes ofpseudodifferential operators preserve these spaces.

Meanwhile, due to the interest of Hardy spaces in applications, various general-izations have been considered such that weighted Hardy spaces [14], Hardy-Lorentzspaces [1, 17] or Hardy-Orlicz type spaces [29]. Recently, the authors [4] introducedgeneralized Hardy spaces Hq(φ) as Hardy spaces modeled in the so-called gener-alized Lorentz spaces �q(φ) (see Sect. 2.2 below). The spaces Hq(φ) include theHardy-Lorentz spaces Hp,q and the classical Hardy spaces Hp as special cases. Sev-eral problems were discussed in [4]. For instance, based on interpolation arguments,a Hardy-Littewood-Sobolev theorem for fractional integrals was established and theboundedness of singular integral operators was studied in the spaces Hq(φ).

In this work we also consider the local counterpart hq(φ) of the generalized Hardyspaces introduced in [4] (cf. Sect. 3). As we will see below, the spaces hq(φ) are moreconvenient for the problems and applications we have in mind. After proving inter-polation results in Sect. 4.1, as simple consequences we study the behavior of theFourier transform on these spaces and show how they are connected with the spacesHq(φ) and �q(φ) (depending on the Boyd indices of φ). The spaces hq(φ) are thenused in Sect. 4.2 to solve a problem on real interpolation of Besov spaces with gener-alized smoothness B

φp,q , which was left open in [13]. With the help of interpolation

tools, we also prove the boundedness of pseudodifferential operators (including theexotic case) both in B

φp,q and in hq(φ) spaces, extending several statements known

for the corresponding classical spaces (see Sect. 5).Finally, in Sect. 6 we give wavelet decompositions for the spaces hq(φ). Close to

this problem, we refer to [10] where the second named author used, for the first time,interpolation tools to obtain subatomic decompositions for spaces of Bessel poten-tial type. More recently, Triebel [45] also used interpolation to get wavelet bases inLorentz spaces. In both papers, the authors have restricted to the Banach case only,and hence linear interpolation was enough. In the present paper, the wavelet represen-tation problem is discussed in the general quasi-Banach setting. This required the useof nonlinear interpolation of vector-valued Lebesgue spaces, so that extra difficultieshad to be circumvented.

2 Preliminaries

Basic Notation By |E| we denote the (Lebesgue) measure of a measurable subsetE of the Euclidean space R

n (n ∈ N). The measure of the unit ball will be simplydenoted by ωn, where we write B(x, r) for the open ball centered at x ∈ R

n and withradius r > 0. By R

n+1+ we denote the space Rn+1+ = {(x, t) : x ∈ R

n, t > 0}. We shalluse the notation 〈x〉 := (1 + |x|2)1/2 for x ∈ R

n.By C∞(Rn) we denote the collection of all infinitely differentiable functions

on Rn. S(Rn) stands for the Schwartz class consisting of all functions from C∞(Rn)

which are rapidly decreasing and S ′(Rn), its topological dual, is the space of all tem-pered distributions. We write f and f ∨, respectively, for the Fourier transform andfor the inverse Fourier transform of f ∈ S(Rn). Both the Fourier transform and itsinverse are extended to S ′(Rn) in the standard way.

J Fourier Anal Appl (2011) 17:691–719 693

The symbol “↪→” means continuous embedding between two quasi-normedspaces. The equivalence f (x) ∼ g(x) (f,g non-negative functions) means that thereare constants c1, c2 > 0 such that c1f (x) ≤ g(x) ≤ c2f (x) for all the admitted val-ues of x. If only one of the above inequalities is intended, then we use the symbol �instead, with an obvious meaning. By C, or c (possibly with subscripts), we denotegeneric positive constants, which may have different values even in the same line.Although the exact values of the constants are usually irrelevant for our purposes,sometimes we emphasize their dependence on certain parameters (e.g. C(p) meansthat C depends on p, etc.). Further notation will be properly introduced wheneverneeded.

2.1 Function Parameters

We are mainly interested in function spaces with generalized parameters taken fromthe so called B class. A function φ : (0,∞) → (0,∞) belongs to the class B if it iscontinuous, φ(1) = 1 and φ(t) := sups>0

φ(st)φ(s)

is finite for every t > 0. For φ ∈ B, the

function φ is submultiplicative and the Boyd lower and upper indices are well-definedby

βφ = limt→0

logφ(t)

log tand αφ = lim

t→+∞logφ(t)

log t,

respectively, with −∞ < βφ ≤ αφ < ∞. Typical functions belonging to B are

φa,b(t) = ta(1 + | log t |)b, a, b ∈ R.

In this case, φa,b(t) = ta(1 + | log t |)|b| and βφa,b= αφa,b

= a.

The main connection between φ ∈ B and φ is made with the estimates

φ(u)

φ(1/t)≤ φ(ut) ≤ φ(u)φ(t), u, t > 0. (2.1)

With the help of (2.1), we may express some properties of φ ∈ B through the Boydindices of φ. For instance,

βφ > 0 if and only if∫ 1

0φ(t)

dt

t< ∞ (2.2)

and

αφ < 0 if and only if∫ ∞

1φ(t)

dt

t< ∞. (2.3)

Furthermore, if βφ > 0 (resp. αφ < 0), then φ is equivalent to some increasing (resp.decreasing) function ψ ∈ B. These (and other) properties of class B may be foundin [27].


2.2 Function Spaces

In general we shall deal with function spaces defined on the whole Rn, so that we

shall omit the Rn in their notation for simplicity.

For 0 < q < ∞ and φ ∈ B, the generalized Lorentz space �q(φ) is defined as thecollection of all measurable functions f on R

n such that

‖f | �q(φ)‖ :=(∫ ∞

0[f ∗(t)φ(t)]q dt

t

)1/q

< ∞, (2.4)

where f ∗ stands for the usual decreasing rearrangement of f (see [6] for details). Onecan show that (2.4) defines a quasi-norm and that �q(φ) is a quasi-Banach space.Clearly, �q(φ) = �q(ψ) if φ(t) ∼ ψ(t).

If φ(t) = t1/p (1 + | log t |)b , with 0 < p < ∞ and b ∈ R, then �q(φ) =Lp,q(logL)b are the Lorentz-Zygmund spaces studied in [5]. They are classicalLorentz spaces Lp,q if b = 0, which in turn are the usual Lebesgue spaces Lp whenq = p.

Generalized Lorentz spaces were considered in [26, 27], for example. They are aparticular case of weighted Lorentz spaces with the weight constructed from class B.A systematic study on general weighted Lorentz spaces can be found in [11], includ-ing further references.

We note that the spaces �q(φ) have the lattice property, that is, for measurablefunctions f,g in R

n it holds

|f | ≤ |g| a.e. ⇒ ‖f | �q(φ)‖ ≤ ‖g | �q(φ)‖.Using the same class B we can consider Besov spaces of generalized smoothness

as follows. Let {ηj }j∈N0 ⊂ S be a system with the following properties:

suppη0 ⊂ {ξ : |ξ | ≤ 2}; suppηj ⊂ {ξ : 2j−1 ≤ |ξ | ≤ 2j+1}, j ∈ N;

supξ∈Rn

|Dβηj (ξ)| ≤ c 2−j |β|, j ∈ N0, β ∈ Nn0, and

∞∑

j=0

ηj (ξ) = 1, ξ ∈ Rn.

For φ ∈ B and 0 < p,q ≤ ∞, the spaces Bφp,q are defined as the collection of all

f ∈ S ′ with the quasi-norm

‖f | Bφp,q‖ := ∥

∥{(ηjf )∨}j∈N0 | �φ

q (Lp)∥

∥, (2.5)

where �φq (E) stands for the space of all sequences a ≡ {aj }j∈N0 from the quasi-

normed space E such that

‖a | �φq (E)‖ :=

( ∞∑

j=0

φ(2j )q‖aj | E‖q

)1/q

(with the usual modification if q = ∞) is finite. The spaces Bφp,q are quasi-Banach

spaces not depending on the system {ηj }j∈N0 chosen (up to the equivalence of quasi-norms).


The spaces Bφp,q , which generalize the classical Besov spaces Bs

p,q , have beenconsidered by several authors from different approaches. We only refer to the paper[15] where historical remarks and further references can be found.

3 Hardy Spaces Modeled on Spaces �q(φ)

In the sequel, ϕ denotes a Schwartz function and ϕt (x) := t−nϕ(x/t), x ∈ Rn, t > 0.

For a tempered distribution f and 0 < σ ≤ ∞, the radial and the non-tangentialmaximal functions are given, respectively, by

(

Rσϕf

)

(x) = sup0<t<σ

|(ϕt ∗ f )(x)| (3.1)

and(

N σϕ f

)

(x) = sup(y,t)∈�σ (x)

|(ϕt ∗ f )(y)|, (3.2)

where �σ (x) := {(y, t) ∈ Rn+1+ : |x − y| < t < σ } denotes the (truncated) cone with

vertex at x ∈ Rn.

The grand maximal function Gσm, m ∈ N0, is defined by

(Gσmf )(x) = sup

�ψ �m≤1(N σ

ψ f )(x) = sup�ψ �m≤1

sup(y,t)∈�σ (x)

|(ψt ∗ f )(y)|, (3.3)

where

�ψ �m := supu∈R

n

|α|≤m

(1 + |u|)m+n|Dαψ(u)|

denotes a suitable collection of seminorms on S .In what follows we assume that our ϕ ∈ S satisfies

∫

Rn ϕ(x) dx = 1 (so that alsoϕt ∈ S with

∫

Rn ϕt (x) dx = 1).The next statement establishes that the three maximal functions above, well-

known in the theory of Hardy spaces, have an equivalent behavior in terms of �q(φ)-quasi-norm.

Theorem 3.1 Let φ ∈ B with βφ > 0, 0 < q < ∞ and 0 < σ ≤ ∞. For a tempereddistribution f , the following conditions are equivalent:

(i) Rσϕf ∈ �q(φ) for some ϕ ∈ S with

∫

Rn ϕ(x) dx = 1.(ii) There is a natural number m0 = m0(n,φ) such that Gσ

mf ∈ �q(φ), for everym ≥ m0.

(iii) N ση f ∈ �q(φ) for some η ∈ S with

∫

Rn η(x) dx = 1. Moreover

∥

∥Rσϕf | �q(φ)

∥

∥ ∼ ∥

∥Gσmf | �q(φ)

∥

∥ ∼ ∥

∥N ση f | �q(φ)

∥

∥

for any f ∈ S ′.


Remark 3.2 Since

Rσψf (x) ≤ N σ

ψ f (x) ≤ CGσmf (x), x ∈ R

n, C ≥ �ψ �m, ψ ∈ S, (3.4)

then the assertions (ii) ⇒ (i) and (ii) ⇒ (iii) are, in fact, valid for all ϕ, η ∈ S underthe assumptions above.

Theorem 3.1 is an extension of maximal characterization results obtained by Fef-ferman and Stein for the spaces Lp = �p(t1/p). It can be proved following the stepsof the proof of Theorem 3.2 in [4] for the case σ = ∞, which in turn were inspired inthe original arguments developed in [16] (see also [24] and [18]). Therefore we justpoint out here what should be modified in Sect. 3.2 of [4] in order that the case offinite σ can be included:

Instead of (N k,Nϕ f )(x), ( Ñ k,N

ϕ f )(x), (T k,N,μϕ f )(x) and (T μ

ϕ f )(x) one should,respectively, consider

(N 1/ε,Nϕ f )(x)

:= sup(y,t)∈�1/ε(x)

|(ϕt ∗ f )(y)|(

t

t + ε − σ−1

)N(

1 + (ε − σ−1)|y|)−N, (3.5)

( Ñ 1/ε,Nϕ f )(x)

:= sup(y,t)∈�1/ε(x)

t |∇y(ϕt ∗ f )(y)|(

t

t + ε − σ−1

)N(

1 + (ε − σ−1)|y|)−N

(3.6)

(T 1/ε,N,μϕ f )(x) := sup

y∈Rn,0<t<1/ε

|(ϕt ∗ f )(y)|(

t

t + ε − σ−1

)N

× (

1 + (ε − σ−1)|y|)−N[

t

|x − y| + t

]μ

,

and

(T σ,μϕ f )(x) := sup

y∈Rn,0<t<σ

|(ϕt ∗ f )(y)|[

t

|x − y| + t

]μ

, (3.7)

where ε ∈ [σ−1,1 + σ−1], N ∈ N0, μ > 0 and ∇y is the usual gradient (with respectto y). Clearly, when ε = σ−1, then N can be taken 0 and (3.5) above then coincideswith (3.2), whereas (3.6) and (3.7) are identical. Moreover, when σ = ∞, the abovefour expressions essentially coincide, respectively, with the expressions (N k,N

ϕ f )(x),

( Ñ k,Nϕ f )(x), (T k,N,μ

ϕ f )(x) and (T μϕ f )(x) of [4] (the difference being that we are

now considering a continuous 1/ε instead of a discrete natural k).Then, with these new expressions, the result corresponding to Lemma 3.5 of [4]

holds as long as ε ∈ (σ−1,1 + σ−1]. Since all the remaining results of that Sect. 4.2can be easily adapted to this more general setting of σ not necessarily infinite, wemay conclude that our Theorem 3.1 above holds true.


It is also not difficult to see that there are essentially only two different cases inthis theorem:

σ = ∞ and σ ∈ (0,∞),

as the following result shows:

Corollary 3.3 Let φ ∈ B with βφ > 0, 0 < q < ∞ and σ, τ ∈ (0,∞). Let ϕ ∈ S with∫

Rn ϕ(x) dx = 1 and f ∈ S ′. Then

Rσϕf ∈ �q(φ) if, and only if, Rτ

ϕf ∈ �q(φ).

Moreover,

‖Rσϕf | �q(φ)‖ ∼ ‖Rτ

ϕf | �q(φ)‖for any f ∈ S ′.

Proof Assume, without loss of generality, that τ > σ . Then

‖Rσϕf | �q(φ)‖ ≤ ‖Rτ

ϕf | �q(φ)‖,since we have (Rσ

ϕf )(x) ≤ (Rτϕf )(x) for all x ∈ R

n.On the other hand, given any x ∈ R

n,

(Rτϕf )(x) = sup

0<s<σ

|((ϕτ/σ )s ∗ f )(x)| = (Rσϕτ/σ

f )(x),

where, as is easily seen, ϕτ/σ ∈ S also satisfies∫

Rn ϕτ/σ (x) dx = 1 (though this isnot really necessary for the conclusion). Therefore, with the help of Theorem 3.1 andRemark 3.2, we get that Rσ

ϕf ∈ �q(φ) implies that Rτϕf ∈ �q(φ) and

‖Rτϕf | �q(φ)‖ ≤ c‖Rσ

ϕf | �q(φ)‖,with c > 0 independent of f . �

For 0 < q < ∞ and φ ∈ B with βφ > 0, we consider then just the two followingtypes of generalized Hardy spaces:

Hq(φ) = {

f ∈ S ′ : ‖f | Hq(φ)‖ϕ := ‖R∞ϕ f | �q(φ)‖ < ∞}

(3.8)

and

hq(φ) = {

f ∈ S ′ : ‖f | hq(φ)‖ϕ := ‖R1ϕf | �q(φ)‖ < ∞}

, (3.9)

where in the latter one we can as well use any other σ ∈ (0,∞) instead of σ = 1, asfollows from the preceding corollary.

A fundamental result of Fefferman and Stein asserts that the definition of Hp doesnot depend on the particular choice of ϕ, in the sense that different choices of ϕ lead toequivalent quasi-norms. Theorem 3.1 and Remark 3.2 show that the same happens inour generalized setting, guaranteeing the consistency of the definitions given in (3.8)and (3.9). By this reason we shall omit the “ϕ” in the notation of the quasi-norms.


We note that the homogeneous space Hq(φ) was recently studied by the authorsin [4].

Examples If φ(t) = t1/p(1 + | log t |)b, 0 < p < ∞, b ∈ R, we would obtain Hardy-Lorentz-Zygmund spaces, which could be denoted by Hp,q(logL)b and hp,q(logL)b ,respectively, following the analogy with the Lorentz-Zygmund spaces. When b = 0they are the so-called Hardy-Lorentz spaces. Their homogeneous version Hp,q wasintroduced in [17] and it was recently studied in [1] (see also [25] for the anisotropiccase). Additionally, if q = p these classes recover the classical real Hardy spaces,namely the homogeneous spaces Hp studied by Fefferman and Stein [16] and thecorresponding local versions hp introduced by Goldberg [18].

The following lemma provides a way of transferring properties from the homoge-neous spaces Hq(φ) to the corresponding local spaces hq(φ). The proof can be doneby adapting to our setting the arguments used to prove Lemma 4 in [18], where theclassical case corresponding to the choice φ(t) = t1/q was dealt with. We sketch ithere, for the sake of completeness.

Lemma 3.4 Let 0 < q < ∞ and βφ > 0. If η ∈ S ,∫

Rn η(x) dx = 1 and∫

Rn xα η(x) dx = 0 for all α �= 0, then there exists c > 0 such that

‖f − η ∗ f | Hq(φ)‖ ≤ c‖f | hq(φ)‖for every f ∈ hq(φ).

Proof Given f ∈ hq(φ), we have that

‖f − η ∗ f | Hq(φ)‖ ≤ ‖ sup0<t<1

|ϕt ∗ f | | �q(φ)‖

+ ‖ sup0<t<1

|(ϕt ∗ η) ∗ f | | �q(φ)‖

+ ‖ supt≥1

|(ϕt − ϕt ∗ η) ∗ f | | �q(φ)‖

= ‖f | hq(φ)‖ + (I ) + (II). (3.10)

In order to deal with the terms (I ) and (II) above, let m ≥ m0(n,φ) be as inTheorem 3.1 and consider σ ∈ (1,∞). Since

‖Gσmf | �q(φ)‖ � ‖f | hq(φ)‖

and

sup�ψ �m≤1

|(ψ ∗ f )(x)| ≤ (Gσmf )(x), ∀x ∈ R

n,

it is enough to show that

∃c > 0 : ∀t ∈ (0,1), �ϕt ∗ η�m ≤ c (3.11)


and

∃c > 0 : ∀t ∈ (1,∞), �ϕt − ϕt ∗ η�m ≤ c. (3.12)

Assertion (3.11) follows from the fact that, for any t ∈ (0,1), u ∈ Rn and α ∈ N

n0 with

|α| ≤ m,

(1 + |u|)m+n|Dα(ϕt ∗ η)(u)|

= (1 + |u|)m+n

∣

∣

∣

∣

∫

Rn

ϕ(z)(Dαη)(u − tz) dz∣

∣

≤∫

Rn

m+n∑

j=0

(

m + n

j

)

(1 + |u − tz|)j |(Dαη)(u − tz)|(1 + |z|)m+n−j |ϕ(z)|dz

≤ c(n,m,ϕ,η),

while in order to prove (3.12) one uses the hypotheses on η and Taylor’s formula withsome ξ ∈ [0,1], allowing us to write that, for any t ∈ (1,∞), u ∈ R

n and α ∈ Nn0 with

|α| ≤ m,

(1 + |u|)m+n|Dα(ϕt − ϕt ∗ η)(u)|

= (1 + |u|)m+nt−n−|α|∣

∣

∣

∣

∫

Rn

[

(Dαϕ)

(

u

t

)

− (Dαϕ)

(

u

t− y

t

)]

η(y)dy

∣

∣

∣

∣

≤ (1 + |u|)m+nt−n−|α|∫

Rn

∑

|β|=m

yβ

β!tm∣

∣

∣

∣

(Dα+βϕ)

(

u − ξy

t

)∣

∣

∣

∣

|η(y)|dy

≤∑

|β|=m

t−n−|α|−m

β!∫

Rn

m+n∑

j=0

(

m + n

j

)

(1 + |u − ξy|)j∣

∣

∣

∣

(Dα+βϕ)

(

u − ξy

t

)∣

∣

∣

∣

× (1 + |y|)2m+n−j |η(y)|dy

≤ c(n,m,ϕ,η). �

Remark 3.5 The hypothesis∫

Rn xα η(x) dx = 0 in the preceding lemma was onlyused in the proof of (3.12) and only for α ∈ N

n0 such that 0 �= |α| < m, where the

integer m has the same meaning as in the proof above.

Concerning the convolution η ∗ f above we would like to make some comments.Using the terminology from [36], we shall say that f ∈ S ′ is bounded if η ∗ f ∈ L∞for every η ∈ S . In [4, Theorem 3.13] we have shown that the elements from Hq(φ),with 0 < q < ∞ and βφ > 0, are bounded distributions. In a similar way, one canshow that this property still holds in hq(φ) (with the same restrictions on q and φ)and

‖η ∗ f | L∞‖ ≤ C‖f | hq(φ)‖,with C > 0 not depending on f .


4 Interpolation with Function Parameter

We recall that, for 0 < p ≤ ∞ and γ ∈ B (with 0 < βγ ≤ αγ < 1), the interpolationspace (A0,A1)γ,p , formed from compatible quasi-normed spaces A0, A1, is the spaceof all a ∈ A0 + A1 such that the quasi-norm

‖a | (A0,A1)γ,p‖ :=(∫ ∞

0

[

γ (t)−1K(t, a)]p dt

t

)1/p

(usual modification if p = ∞) is finite, where

K(t, a) = K(t, a;A0,A1) = infa0+a1=a

a0∈A0,a1∈A1

(‖a0|A0‖ + t‖a1|A1‖), t > 0,

is the well-known Peetre K-functional. The space (A0,A1)γ,p is continuously em-bedded between A0 ∩ A1 and A0 + A1. As usual, if γ (t) = tθ then we simply write(A0,A1)θ,p to denote the corresponding interpolation space. We refer to [7, 27, 41]for further information on real interpolation.

The next statement shows how the interpolation of operators not necessarily linearcan be dealt with.

Lemma 4.1 Let (A0,A1) and (B0,B1) be compatible couples of quasi-normedspaces. Let T : A0 + A1 −→ B0 + B1 be a (possibly nonlinear) operator. Supposethat there are constants Mi > 0 such that, for each couple (a0, a1), with ai ∈ Ai ,there exist bi ∈ Bi satisfying

T (a0 + a1) = b0 + b1 and ‖bi | Bi‖ ≤ Mi‖ai | Ai‖, i = 0,1.

If 0 < q < ∞ and γ ∈ B with 0 < βγ ≤ αγ < 1, then there holds

‖T a | (B0,B1)γ,q‖ ≤ M0γ (M1/M0)‖a | (A0,A1)γ,q‖ (4.1)

for every a ∈ (A0,A1)γ,q .

We note that this result is formulated in [34, Theorem 3] without proof, in the caseof interpolation with power parameter. For the sake of completeness we shall give aproof later in the Appendix for the general case.

4.1 Interpolation of Generalized Hardy Spaces

Although all the interpolation formulas below involving the homogeneous spacesHq(φ) have been proved in [4], we recall them here for completeness, showing at thesame time the parallelism with the local counterpart.

Theorem 4.2 Let 0 < r < q < ∞ and γ ∈ B with 0 < βγ ≤ αγ < 1. Then

(Hr,L∞)γ,q = Hq(φr), (4.2)

(hr ,L∞)γ,q = hq(φr), (4.3)


where φr(t) := t1/r

γ (t1/r ).

We will prove this theorem later on, as we want to give first some important con-sequences.

Using reiteration arguments (cf. [4, 27]), from Theorem 4.2 we obtain the follow-ing interpolation formulas.

Theorem 4.3 Let 0 < q0, q1, q < ∞, φ0, φ1, γ ∈ B and ψ = φ0/φ1. If βφi> 0 (i =

0,1), 0 < βγ ≤ αγ < 1 and βψ > 0 or αψ < 0, then

(

Hq0(φ0),Hq1(φ1))

γ,q= Hq(φ), (4.4)

(

hq0(φ0), hq1(φ1))

γ,q= hq(φ), (4.5)

where φ(t) = φ0(t)γ (ψ(t))

.

Remark 4.4 For φi(t) = t1/pi , i = 0,1, and γ (t) = tθ (4.4) gives the former state-ment obtained by Fefferman, Riviere and Sagher [17] for Hardy-Lorentz spaces,namely if 0 < θ < 1, 0 < q0, q1, q < ∞, 0 < p0 �= p1 < ∞, then there holds

(

Hp0,q0 ,Hp1,q1

)

θ,q= Hp,q,

1

p= 1 − θ

p0+ θ

p1.

In particular,

(Hp0 ,Hp1)θ,p = Hp.

Similarly (4.5) generalizes the classical local counterpart

(hp0 , hp1)θ,p = hp, 0 < p0,p1 < ∞,1

p= 1 − θ

p0+ θ

p1

also for 0 < p0 �= p1 < ∞, 1p

= 1−θp0

+ θp1

(see [39, Sect. 2.4] for further details).

From Theorem 4.3 one can see that the generalized Hardy spaces Hq(φ) and hq(φ)

can be written as interpolation spaces between their classical analogues.

Corollary 4.5 Let 0 < q < ∞ and φ ∈ B with βφ > 0. If p0,p1 satisfy 0 < 1p1

<

βφ ≤ αφ < 1p0

< ∞, then

Hq(φ) = (Hp0 ,Hp1)η,q (4.6)

and

hq(φ) = (hp0 , hp1)η,q , (4.7)

where

η(t) = tp1

p1−p0

φ(tp0p1

p1−p0 )

. (4.8)


In order to prove Theorem 4.2 for the local Hardy spaces, we need a result some-what similar to [4, Lemma 4.2], which was used in the proof for the homogeneouscounterpart. This is a decomposition of a special type of function into “good” and“bad” parts. Though the outcome is similar to [4, Lemma 4.2], the proof is quite dif-ferent, following closely the ideas in [8, pp. 594–595], instead of [17], where theproof of the homogeneous counterpart was based on. In fact, the latter approachwould break down in the case of local Hardy spaces. For the convenience of thereader, we will make a brief sketch here.

Lemma 4.6 Let f ∈ S ′ and ψ ∈ S be such that ψ �= 0 and ψ ∗ f ∈ L∞ ∩ C∞(of course, only the inclusion in L∞ is an extra requirement here). Let 0 < p < ∞,δ ≥ 0, σ > 2 and m ∈ N0. It is possible to write ψ ∗ f = g + b, where the tempereddistributions g and b satisfy

‖g | L∞‖ ≤ c1δ and ‖b | hp‖p ≤ c2

∫

{Gσmf >δ}

(Gσmf )p(x) dx,

where c1, c2 > 0 are independent of f,g, b, δ and σ .

Proof Consider a tessellation {Ik}k∈N of Rn by disjoint (half-closed) unit cubes with

sides parallel to the coordinate axes.Given f and ψ as in the statement of the lemma, consider, for each k ∈ N, f k :=

(ψ ∗ f )χIk. Notice that

∑∞k=1 f k = ψ ∗ f both pointwise and in S ′.

Consider E1 := {k ∈ N : ‖f k|L∞‖ ≤ c1δ}, with c1 to be fixed later on (in depen-dence of n,m and ψ only), and E2 := N \ E1 and define

g :=∑

k∈E1

f k and b := ψ ∗ f − g =∑

k∈E2

f k.

It is immediate that ‖g|L∞‖ ≤ c1δ.On the other hand, fixing (for the calculations with ‖ · |hp‖) a ϕ ∈ S with

∫

Rn ϕ(x) dx = 1 such that ϕ ≥ 0 and with support in the open unit cube centred at theorigin, denoting by I ∗

k the closed cube with the same centre as Ik but with the (paral-lel) sides of length 2, and writing �∗ := ⋃

k∈E2I ∗k and E2,j := {k ∈ E2 : I ∗

k ∩I ∗j �= ∅},

j ∈ E2, we get that

‖b|hp‖p = ‖ sup0<t<1

|ϕt ∗ b||Lp‖ =∫

�∗sup

0<t<1

∣

∣

∣

∣

(

ϕt ∗∑

k∈E2

f k

)

(x)

∣

∣

∣

∣

p

dx

≤∑

j∈E2

∫

I∗j

sup0<t<1

(

ϕt ∗∑

k∈E2,j

|f k|)p

(x) dx

�∑

j∈E2

∑

k∈E2,j

‖f k|L∞‖p �∑

k∈E2

‖f k|L∞‖p. (4.9)


Moreover, defining η(x) := 2nψ(2x), x ∈ Rn, and c1 := �η�m, we have that, for any

x ∈ Ik ,

‖f k|L∞‖ = supy∈Ik

|(η2 ∗ f )(y)| ≤ sup(y,t)∈�σ (x)

|(ηt ∗ f )(y)| ≤ c1(Gσmf )(x).

Using this in (4.9), we finally arrive at

‖b|hp‖p �∑

k∈E2

∫

Ik

‖f k|L∞‖p dx ≤∫

{Gσmf >δ}

(Gσmf )p(x) dx.

�

Proof of Theorem 4.2 Formula (4.2) is given by Theorem 4.3 in [4], so that only (4.3)needs to be proven. As in the homogeneous case, the embedding (hr ,L∞)γ,q ↪→hq(φr) follows from the interpolation formula (Lr,L∞)γ,q = �q(φr) (see [27, The-orem 3]) and Lemma 4.1, by observing that R1

ϕ is a sublinear operator which isbounded from hr into Lr and from L∞ into itself. The use of Lemma 4.1 can bedone just as in the proof of Lemma 6.4 below, with natural adaptations.

For the converse embedding, we make use of some arguments from [8, 17] (seealso [4]). Let f ∈ hq(φr). Take ψ ∈ S with

∫

Rn ψ(x)dx = 1 and∫

Rn xα ψ(x)dx = 0for all α �= 0.

By Lemma 3.4 and formula (4.2) we have

f − ψ ∗ f ∈ Hq(φr) = (Hr,L∞)γ,q ↪→ (hr ,L∞)γ,q . (4.10)

As to ψ ∗ f , since f is a bounded distribution we know that ψ ∗ f ∈ L∞ ∩ C∞,hence Lemma 4.6 can be applied with p = r and σ > 2: for any t > 0 and m ∈ N0,there are tempered distributions gt and bt such that gt + bt = ψ ∗ f , with

‖gt | L∞‖ ≤ c1(Gσmf )∗(tr ) and ‖bt | hr‖r ≤ c2

∫

�t

(Gσmf (x))rdx,

where �t := {x ∈ Rn : Gσ

mf (x) > (Gσmf )∗(tr )} and c1, c2 > 0 are independent of

t, gt , bt and f . Proceeding now as in [4, proof of Theorem 4.3], with a sufficientlylarge m in order that ‖Gσ

mf | �q(φr)‖ < ∞ (cf. (3.9), Corollary 3.3 and Theo-rem 3.1), one can show that

‖ψ ∗ f | (hr ,L∞)γ,q‖ ≤ c‖Gσmf | �q(φr)‖ ∼ ‖f | hq(φr)‖.

From this, (4.10) and Lemma 3.4 we can complete the proof:

‖f | (hr ,L∞)γ,q‖ � ‖f − ψ ∗ f | (hr ,L∞)γ,q‖ + ‖ψ ∗ f | (hr ,L∞)γ,q‖� ‖f | hq(φr)‖. �

Now we give two properties of the spaces hq(φ) which generalize important factsknown for the classical Hardy spaces hp .


Connection Between Hq(φ), hq(φ) and �q(φ) It is known that both the clas-sical Hardy spaces Hp = Hp(t1/p) and hp = hp(t1/p) coincide with the usualLebesgue spaces Lp when 1 < p < ∞. In general, from the definition, we haveHq(φ) ↪→ hq(φ). However, using interpolation tools we can get more informationwithin this generalized context. We note that in [4] it was shown that Hq(φ) = �q(φ)

if 0 < q < ∞ and 0 < βφ ≤ αφ < 1. For completeness we describe here how to get a

similar result for the local spaces. Let us take p0,p1 such that 0 < 1p1

< βφ ≤ αφ <

1p0

< 1. Since 1 < p0, p1 < ∞ then Hpi= hpi

= Lpi, i = 0,1. Combining this with

Corollary 4.5 and the corresponding result for Lorentz spaces (cf. [27], Theorem 3),we get

hq(φ) = (hp0 , hp1)η,q = (Lp0 ,Lp1)η,q = �q(φ),

where η is the function from (4.8). Thus the following result holds.

Theorem 4.7 Let 0 < q < ∞, φ ∈ B with 0 < βφ ≤ αφ < 1. Then, up to equivalenceof quasi-norms, we have

Hq(φ) = hq(φ) = �q(φ).

The Fourier Transform In [4] we have shown that the Fourier transform of f ∈Hq(φ) is a continuous function on R

n and behaves like

| f (ξ)| ≤ c|ξ |−n φ(|ξ |n)‖f | Hq(φ)‖, 0 < q < ∞, βφ > 1, (4.11)

with c > 0 independent of ξ ∈ Rn and f ∈ Hq(φ) (interpreting |ξ |−n φ(|ξ |n) = 0

when ξ = 0). Estimate (4.11) was obtained from its classical analogue by means ofappropriate interpolation techniques.

As regards local Hardy spaces hp , 0 < p ≤ 1, it is known that there exists c > 0such that

| f (ξ)| ≤ c〈ξ 〉n(1/p−1)‖f | hp‖ (4.12)

for every ξ ∈ Rn and f ∈ hp . A proof of (4.12), based on atomic decompositions,

can be found in [23].Following the steps of the proof of Theorem 5.1 in [4], we can establish the es-

timate below, which gives an extension of (4.12) to the generalized Hardy spaceshq(φ).

Proposition 4.8 Let 0 < q < ∞ and φ ∈ B. If f ∈ hq(φ) and βφ > 1, then f iscontinuous on R

n and

| f (ξ)| ≤ c〈ξ 〉−nφ(〈ξ 〉n)‖f | hq(φ)‖,

where c > 0 is independent of ξ ∈ Rn and f ∈ hq(φ).


4.2 Interpolation of Generalized Besov Spaces

For φ0, φ1, γ ∈ B and 0 < p,q0, q1, q ≤ ∞, we have(

Bφ0p,q0

,Bφ1p,q1

)

γ,q= Bφ

p,q, (4.13)

where 0 < βγ ≤ αγ < 1, β(

φ0φ1

)> 0 (or α

(φ0φ1

)< 0), and φ(t) := φ0(t)

γ (φ0(t)

φ1(t)). This state-

ment was proved by Cobos and Fernandez [13] in the Banach case p,q ≥ 1 by usingthe so-called retraction and co-retraction method (see, for example, [7, 41] for detailson this interpolation method), which allows reduction of the problem to the interpo-lation of appropriate sequence spaces. In fact, they observed that B

φpq is a retract of

�φq (Lp), where

R{fj }j∈N0 :=∞∑

j=0

(ηjfj )

∨, with ηj =1

∑

r=−1

ηj+r (ϕ−1 ≡ 0), (4.14)

(convergence in S ′) is a retraction from �φq (Lp) to B

φpq and

Jf := {(ηjf )∨}j∈N0 (4.15)

is the corresponding co-retraction from Bφpq to �

φq (Lp). Here {ηj }j∈N0 is a system

with the properties described in Sect. 2.2. The quasi-Banach case can also be treatedthrough this method. However, since the mapping R above is meaningless for 0 <

p < 1, the space Lp needs to be replaced by the Hardy space hp for such values of p

(cf. Remark 5.4 in [13]). A detailed description how this question can be dealt within the general case can be found in [2].

Formula (4.13) shows that real interpolation between generalized Besov spacesB

φpq(Rn) with p fixed produces a space of the same type. However, a change of the

parameter p leads to different spaces, which are not included in the scale defined by(2.5). Indeed, even in the classical situation there holds

(Bsp0,q

,Bsp1,q

)θ,q = Bsp,q(q), (4.16)

where 1p

= 1−θp0

+ θp1

, with 1 < p0 �= p1 < ∞, 1 ≤ q < ∞ and 0 < θ < 1 (cf. [41],Theorem 2.4.1). Here Bs

p,q(q) denotes the space obtained as in (2.5) (with φ(t) = t s ),but replacing the space Lp by the Lorentz space Lp,q .

The generalization of (4.16) to the spaces Bφpq was also obtained in [13] in the

Banach case only, by using the already mentioned retraction method with the samemappings (4.14) and (4.15): for 1 ≤ p0 �= p1 ≤ ∞, 1 ≤ q < ∞ and 0 < βγ ≤ αγ < 1,Theorem 5.8 in [13] establishes that

(

Bφp0,q

,Bφp1,q

)

γ,q= Bφ

ρ,q, ρ(t) = t1/p0γ (t1/p0−1/p1)−1,

where Bφρ,q is a certain Besov type space modeled on the generalized Lorentz space

�q(ρ) instead of the usual Lp-space.


Nevertheless, the quasi-Banach case (i.e. when the parameters p,q are allowedto be less than one) was not investigated in [13] and, to the best of our knowledge,has remained open. Recently, in [3] an approach based on wavelet decompositionsobtained in [2] was considered to construct a new retraction with the advantage ofworking for the full range of the parameters. This method proved to give informa-tion about the interpolation spaces for all cases, but, unfortunately, it produced exactinterpolation formulas only for power interpolation parameters γ (t) = tθ .

Now we have the right tools to solve the general case at once. To this end let usdefine new generalized spaces of Besov type.

Definition 4.9 Let 0 < q, r < ∞ and φ,ψ ∈ B with βψ > 0. We define Bφψ,q,r as the

space of all those f ∈ S ′ such that

∥

∥f | Bφψ,q,r

∥

∥ := ∥

∥{(ηjf )∨}j∈N0 | �φ

q (hr(ψ))∥

∥ < ∞.

If ψ(t) = t1/p and r = p then Bφψ,q,r = B

φp,q are the generalized Besov spaces

mentioned above, which in turn coincide with the classical Besov spaces Bsp,q when

φ(t) = t s . We also note that for φ(t) = t s and ψ(t) = t1/p then Bφψ,q,r = Bs

p,q,(r) arethe Besov-type spaces considered in [41, Sect. 2.4], which were specifically intro-duced to describe interpolation spaces.

The next statement extends [13, Theorem 5.8] to the quasi-Banach case.

Theorem 4.10 Let 0 < p0 �= p1 < ∞, 0 < q < ∞ and φ,γ ∈ B with 0 < βγ ≤ αγ <

1. Then(

Bφp0,q

,Bφp1,q

)

γ,q= B

φψ,q,q ,

where

ψ(t) := t1/p0

γ (t1/p0−1/p1). (4.17)

Proof First we observe that Bφp,q is a retract of �

φq (hp) with respect to the retraction

and co-retraction defined above (cf. [2, 13]). Moreover, we have

(

�φq (hp0), �

φq (hp1)

)

γ,q= �φ

q ((hp0 , hp1)γ,q)

(see [33], Proposition 3.2). Therefore, we get successively

∥

∥f | (Bφp0,q

,Bφp1,q

)γ,q

∥

∥ ∼ ∥

∥Jf | (�φq (hp0), �

φq (hp1)

)

γ,q

∥

∥

∼ ∥

∥Jf | �φq ((hp0 , hp1)γ,q)

∥

∥

∼ ∥

∥Jf | �φq ((hq(ψ))

∥

∥,

where in the last equivalence we used formula (4.5) with ψ given by (4.17). �


5 Applications to Pseudodifferential Operators

Pseudodifferential operators are widely considered in the literature. We only refer tosome standard references [22, 37, 38, 40] and the references therein for the basic factson this subject.

We will be interested in operators whose symbols belong to the Hörmanderclass S

μ1,δ , μ ∈ R and 0 ≤ δ ≤ 1, consisting of those complex-valued C∞ func-

tions a = a(x, ξ) in Rn × R

n such that for all multi-indices α = (α1, . . . , αn),β =(β1, . . . , βn) ∈ N

n0 there exists a constant cα,β > 0 with

∣

∣Dαx D

βξ a(x, ξ)

∣

∣ ≤ cα,β 〈ξ 〉μ−|β|+δ|α|

for all x, ξ ∈ Rn, where we use the abbreviation

Dαx a(x, ξ) = ∂ |α|a(x, ξ)

∂xα11 · · · ∂x

αnn

and Dβξ a(x, ξ) = ∂ |β|a(x, ξ)

∂ξβ11 · · · ∂ξ

βnn

.

The pseudodifferential operator a(x,D) associated to the symbol a ∈ Sμ1,δ is for-

mally defined by

a(x,D)f (x) =∫

Rn

eixξ a(x, ξ) f (ξ) dξ, x ∈ Rn, f ∈ S

(further details can be found in the references above).As is known, pseudodifferential operators with symbols in the class S0

1,δ arebounded in Lp , 1 < p < ∞, when δ < 1 (cf. [22, 38]). However, this property doesnot hold in general when the symbols are taken from the exotic class S0

1,1. For in-stance, Ching [12] gave an example of an operator with a symbol in this class whichis not bounded in L2.

Our aim below is to show how the interpolation properties discussed in Sect. 4can be used to study the action of pseudodifferential operators on the generalizedBesov and Hardy spaces considered above. The exotic case corresponding to the caseδ = 1 will be also included in this discussion due to its strong interest raised in theliterature.

5.1 Pseudodifferential Operators in Besov Spaces

Pseudodifferential operators with symbols belonging to Sμ1,δ , μ ∈ R, 0 ≤ δ < 1, are

bounded from Bs+μp,q into Bs

p,q , for all 0 < p,q ≤ ∞ and s ∈ R. Except for somelimiting cases of p and q , this assertion was shown by Päivärinta [30] (see also [40],Chap. 6).

These mapping properties are not assured in general when we deal with exoticsymbols. Nevertheless, as shown by Runst [35], the boundedness assertion holdingfor symbols in S

μ1,δ , 0 ≤ δ < 1, can be recovered in the exotic case when sufficiently

large smoothness is required to the space. More precisely, if

s > n

(

1

min(1,p)− 1

)

(5.1)


then pseudodifferential operators with symbols in Sμ1,1 map continuously B

s+μp,q into

Bsp,q . Other references and details about pseudodifferential operators with exotic

symbols can be found in [40].

Remark 5.1 For future reference we note that the behavior of pseudodifferential op-erators on the Triebel-Lizorkin scale F s

p,q (with p < ∞) is similar to that alreadydescribed for the Besov spaces. However, when dealing with the exotic class S

μ1,1,

the restriction corresponding to (5.1) needs to be replaced by s > n( 1min(1,p,q)

− 1).The proofs of these facts can be found in the references mentioned above. We alsorefer to [19], where the extension of these assertions to weighted spaces was given.

Now we are interested in the behavior of pseudodifferential operators acting ongeneralized Besov spaces. First we note that every space B

φp,q can be written as an

appropriate interpolation space between classical Besov spaces. In fact, given 0 <

p,q ≤ ∞ and φ ∈ B, from (4.13) we may write

Bφp,q = (

Bs0p,q0

,Bs1p,q1

)

γ,q

for any 0 < q0, q1 ≤ ∞ and s0, s1 ∈ R such that s1 < βφ ≤ αφ < s0, with γ ∈ B given

by γ (t) = t

s0s0−s1

φ(t1

s0−s1 )

. Then from formula (4.13) we can also easily derive the following

criterion.

Lemma 5.2 Let 0 < p,q0, q1, r0, r1 ≤ ∞ and s0, s1, σ0, σ1 ∈ R with s0 > s1 andσ0 �= σ1. Suppose that T is a linear operator from B

s0p,q0 + B

s1p,q1 into B

σ0p,r0 + B

σ1p,r1

which is bounded from Bsip,qi

into Bσip,ri , i = 0,1. Then, for every 0 < q ≤ ∞ and

φ ∈ B such that s1 < βφ ≤ αφ < s0, T is bounded from Bφp,q into B

ψp,q , where

ψ(t) = tσ0−κs0 φ(tκ), κ := σ0 − σ1

s0 − s1.

Using this result we get the following statement on the continuity of pseudodiffer-ential operators on spaces B

φp,q .

Theorem 5.3 Let φ ∈ B and 0 < p,q ≤ ∞. Let also μ ∈ R and 0 ≤ δ < 1. If a ∈ Sμ1,δ

then the operator a(x,D) is bounded from Bφp,q into B

ψp,q , where ψ ∈ B is defined

by

ψ(t) = t−μ φ(t). (5.2)

Proof We choose s0, s1 such that s1 < βφ ≤ αφ < s0. Since a(·,D) is bounded from

Bsip,q into B

si−μp,q , i = 0,1, then the result follows from Lemma 5.2 (with σi = si − μ

and hence κ = 1). �

For pseudodifferential operators with symbols belonging to the exotic class Sμ1,1,

we have the following result.


Theorem 5.4 Let 0 < p,q ≤ ∞ and μ ∈ R. If a ∈ Sμ1,1 then the operator a(x,D) is

bounded from Bφp,q into B

ψp,q for every φ ∈ B such that

βφ > μ + n

(

1

min(1,p)− 1

)

, (5.3)

where ψ is given by (5.2).

Proof The proof is similar to the previous one. We only have to choose s0, s1 suchthat s0 > αφ ≥ βφ > s1 > μ + n( 1

min(1,p)− 1). �

Remark 5.5 We restricted ourselves to the study of pseudodifferential operators withsymbols in the regular Hörmander classes. However, we point out that the same inter-polation tools could also be used to obtain results on the continuity of pseudodiffer-ential operators on spaces B

φp,q whose symbols belong to other classes. For instance,

see [8, 9], where Besov type classes of symbols were considered.

5.2 Pseudodifferential Operators in Hardy Spaces

In this section we discuss the continuity of pseudodifferential operators acting onlocal Hardy spaces. The first step in this direction was done by D. Goldberg. Usingatomic decompositions, in [18] he showed that hp , 0 < p < ∞, is preserved by theseoperators when their symbols belong to class S0

1,0. However, since hp = F 0p,2 for

0 < p < ∞, from the comments made in Remark 5.1 one knows, more generally,that the operators a(x,D) map continuously hp into itself when a ∈ S0

1,δ , 0 ≤ δ < 1.Further results on the continuity of pseudodifferential operators in these spaces canbe found in [31, 32, 38].

Recently J. Hounie and R. Kapp [23] carried out a more systematic study on theaction of pseudodifferential operators on hp spaces, with applications to the solvabil-ity of planar vector fields (see also [21]). Naturally, one may ask now for the behaviorof such operators on the generalized Hardy spaces defined in (3.9).

Following the spirit of Lemma 5.2, we can obtain the next statement from Theo-rem 4.3 and Corollary 4.5.

Lemma 5.6 Let 0 < p0 < p1 < ∞, 0 < q0 �= q1 < ∞ and ϑ := 1/q0−1/q11/p0−1/p1

. Supposethat T is a linear operator from hp0 + hp1 into hq0 + hq1 which is bounded from hpi

into hqi, i = 0,1. Then

T is bounded from hq(φ) into hq

(

t1q0

−ϑ 1p0 φ(tϑ )

)

for every 0 < q < ∞ and φ ∈ B such that 1p1

< βφ ≤ αφ < 1p0

.

Taking into account this interpolation argument, one can prove the following:

Theorem 5.7 Let 0 < q < ∞ and φ ∈ B with βφ > 0. If a ∈ S01,δ , 0 ≤ δ < 1, then the

pseudodifferential operator a(·,D) is bounded from hq(φ) into itself.


When the symbols are taken from the exotic class S01,1 then the corresponding

pseudodifferential operators do not necessarily map hp , 0 < p < ∞, into itself (cf.[12] for the case h2 = L2). Due to this fact, several subclasses of S0

1,1 have beenconsidered in order to give rise to pseudodifferential operators continuous in thesespaces. An example is the Bony class BS0

1,1 consisting of all those symbols a ∈ S01,1

having certain restrictions on the support of the partial Fourier transform of a(x, ξ)

(with respect to the first variable), see [23, 38] for details.Pseudodifferential operators with such symbols are bounded in Lp , 1 < p < ∞.

In [23] the authors proved that the same assertion holds for the hp spaces with 0 <

p ≤ 1. Hence, using Lemma 5.6 again, we can extend these arguments to our gener-alized setting.

Theorem 5.8 Let 0 < q < ∞ and φ ∈ B with βφ > 0. Then the pseudodifferential

operators with symbols belonging to class BS01,1 are bounded in hq(φ).

6 Wavelet Decompositions of hq(φ)

This section deals with discrete representations of the Hardy spaces hq(φ).

6.1 Vector-Valued Spaces

In the sequel we shall make use of a vector-valued version of Lorentz spaces. For0 < q < ∞ and φ ∈ B, one defines the generalized vector-valued Lorentz space�q(φ, �2) as the class of all strongly measurable functions f : R

n → �2 such that‖f | �q(φ, �2)‖ < ∞, where the quasi-norm ‖· | �q(φ, �2)‖ is given by (2.4) withthe decreasing rearrangement interpreted in the vector sense, i.e. f ∗ = ‖f (·) | �2‖∗(see [7, 10, 41]).

Before passing to the discrete decompositions of Hardy spaces, we discusssome useful interpolation tools. First of all, we note that all the spaces Lr(�2),0 < r ≤ ∞, are continuously embedded into the Hausdorff topological vector space(t.v.s.) M(Rn, �2) of all strongly measurable functions (from R

n into �2), endowedwith the metric of convergence in measure.

Given 0 < r < q < ∞ and γ ∈ B with 0 < βγ ≤ αγ < 1, then formula

(Lr(�2),L∞(�2))γ,q = �q(φr , �2) (6.1)

holds, where φr(t) = t1/r

γ (t1/r ). As in the scalar case, this can be shown using the well-

known estimate of the K-functional in terms of the decreasing rearrangement of f ,which still holds in the vector case (see [41, p. 135] for details and references), i.e.

K(t, f ;Lr(�2),L∞(�2)) ∼(∫ t r

0f ∗(s)r ds

)1/r

, 0 < r < ∞.

Using formula (6.1) and reiteration [27, Theorem 2], we get the formula

(�q0(φ0, �2),�q1(φ1, �2))γ,q = �q(φ, �2), with φ = φ0

γ ◦ (φ0/φ1),


which holds under the same assumptions of Theorem 4.3. Consequently, for any 0 <

q < ∞ and φ ∈ B, if 0 < 1p1

< βφ ≤ αφ < 1p0

< ∞ then we have

�q(φ, �2) = (Lp0(�2),Lp1(�2))η,q , (6.2)

where η is the function given by (4.8).

6.2 Wavelet Representations

We shall deal with real compactly supported wavelets of Daubechies (inhomoge-neous) type. A systematic study on the construction of wavelet bases in functionspaces, including various references, can be found in the recent monographs [43, 44].

Let �k , k ∈ N, be the system {ψj,Gm }(j,G,m) consisting of real compactly supported

wavelets in Ck , with vanishing moments until order k, given by

ψj,Gm (x) =

{

ψGm (x), j = 0, G ∈ G0, m ∈ Z

n,

2j−1

2 nψGm (2j−1x), j ∈ N, G ∈ Gj, m ∈ Z

n,(6.3)

where Gj denotes sets of n-tuples which count the possible combinations of basicfather and mother wavelets to be considered. This follows essentially the notationfrom [43, Chaps. 1, 3], and we do not go into further details. For simplicity, we shallomit the set of indices from the notation.

It is known that the system �k (with k large enough) provides unconditionalSchauder bases in many classical function spaces, such as Lebesgue spaces, Besselpotential spaces, Hardy spaces and Hölder-Zygmund spaces (see, for instance, [28,Chap. 6]). Based on atomic decompositions, local means and duality theory, Triebel[42] has extended these assertions to the entire scales of Besov and Triebel-Lizorkinspaces. Later Triebel and Haroske [20] dealt also with the weighted case. Using in-terpolation techniques, more recently it was shown in [2] that the same system alsogives unconditional bases in the generalized Besov spaces B

φp,q .

Let λp , 0 < p < ∞, be the space of all sequences μ = {μj,Gm } ⊂ C for which the

quasi-norm

‖μ | λp‖ :=∥

∥

∥

∥

(

∑

j,G,m

|μj,Gm χjm(·)|2

)1/2 ∣

∣

∣

∣

Lp

∥

∥

∥

∥

(6.4)

is finite, where χjm denotes the characteristic function of the cube Qjm in Rn, with

sides parallel to the axes, centred at 2−jm and with side length 2−j+1, where j ∈ N0and m ∈ Z

n.The next statement is contained in [43, Theorem 3.5, see also footnote in p. 156].

Proposition 6.1 Let 0 < p < ∞ and let �k be the wavelet system above with N �k > n( 1

min(p,2)− 1)+. Then f ∈ S ′ belongs to hp if, and only if, it can be represented

as

f =∑

j,G,m

μj,Gm 2−jn/2ψ

j,Gm with μ = {μj,G

m } ∈ λp, (6.5)


(summability in S ′). Moreover, the wavelet coefficients μj,Gm are uniquely determined

by

μj,Gm = μ

j,Gm (f ) := 2jn/2〈f,ψ

j,Gm 〉. (6.6)

Further, f �→ {2jn/2〈f, ψj,Gm 〉} is an isomorphic map of hp onto λp , and �k is an

unconditional basis in hp .

Remark 6.2 The assumption on k assures, in particular, that the duality appearing in(6.6) makes sense, since then the wavelets ψ

j,Gm belong to the dual space of hp . Note

also that the statement above provides an equivalent quasi-norm in hp:

‖f | hp‖ ∼ ‖μ(f ) | λp‖, μ(f ) ≡ {μj,Gm (f )}.

Given 0 < q < ∞ and φ ∈ B, let us denote by λq(φ) the vector space

λq(φ) := {

μ ≡ {μj,Gm } ⊂ C : ‖μ | λq(φ)‖ < ∞}

,

where the quasi-norm ‖· | λq(φ)‖ is defined as in (6.4) but with �q(φ) in place of theLp space. Of course, in particular, we have λp(t1/p) = λp , 0 < p < ∞.

As in [10], it can be shown that λq(φ) can be equivalently described as the space

of all μ ≡ {μj,Gm } ⊂ C such that x �→ {μj,G

m χjm(x)} belongs to �q(φ, �2).We want to show that the sequence space λq(φ) is an interpolation space between

λp-spaces. First we give an auxiliary result whose proof will be given in the Appen-dix.

Lemma 6.3 Let I be any set and let A0 and A1 be any quasi-normed spaces of com-plex functions defined on I (if I is countable, then these are, in fact, sequence spaces).Assume that both A0 and A1 are continuously embedded in some fixed Hausdorff t.v.s.formed by complex functions on I , so that, in particular, (A0,A1) is a compatible pairfor interpolation.

If, for each k ∈ {0,1},|f | ≤ |g| ∧ g ∈ Ak ⇒ f ∈ Ak ∧ ‖f |Ak‖ � ‖g|Ak‖, (6.7)

then the same happens with (A0,A1)γ,q instead of Ak , for any given γ ∈ B and0 < q ≤ ∞.

Lemma 6.4 Let 0 < q < ∞ and φ ∈ B with βφ > 0. For any p0,p1 satisfying 0 <1p1

< βφ ≤ αφ < 1p0

< ∞, there holds

λq(φ) = (λp0 , λp1)η,q ,

with η defined by (4.8).

Proof Taking into account the characterization of each λpkmentioned above, it is

clear that S : λp0 + λp1 −→ Lp0(�2) + Lp1(�2), given by

Sμ = [

x �→ {

μj,Gm χjm(x)

}]

, μ ≡ {

μj,Gm

}

,


is a well-defined linear operator. Moreover, its restriction to λpkis a bounded opera-

tor into Lpk(�2), k = 0,1. Therefore, the interpolation property combined with (6.2)

show that [x �→ {μj,Gm χjm(x)}] ∈ �q(φ, �2) if μ ∈ (λp0, λp1)η,q and

‖μ | λq(φ)‖ � ‖μ | (λp0 , λp1)η,q‖.Hence, we conclude that (λp0, λp1)η,q ↪→ λq(φ).

For the converse embedding, we make use of the following (nonlinear) operator,for some given positive r :

R[

x �→ {

aj,Gm (x)

}] :={(

1

|Qjm|∫

Qjm

|aj,Gm (y)|r dy

)1/r}

.

We want to show that, for some suitable r > 0, R : Lp0(�2)+Lp1(�2) −→ λp0 + λp1

and that its restriction to (Lp0(�2),Lp1(�2))η,q acts boundedly into (λp0 , λp1)η,q .

First observe that the expression∫

Qjm|aj,G

m (y)|rdy makes sense for every

(j,G,m) if we take 0 < r ≤ p0(< p1). For ak ∈ Lpk(�2) (k = 0,1), we have

Ri(a0 + a1) ≤ c(r) (Ri(a

0) + Ri(a1)) for all i, where we write {Rig} for the se-

quence Rg. Define

μki := Ri(a

0 + a1)

Ri(a0) + Ri(a1)Ri(a

k) (with μki = 0 if Ri(a

0) + Ri(a1) = 0).

Then μ0 + μ1 = R(a0 + a1). Since ‖μk | λpk‖ ≤ c(r)‖R(ak) | λpk

‖, then the claimfollows from Lemma 4.1 provided one shows that the restriction of R to each Lpk

(�2)

is a bounded operator into λpk. We prove this next:

∥

∥R[

x �→ {

aj,Gm (x)

}] | λpk

∥

∥

=∥

∥

∥

∥

(

∑

j,G,m

∣

∣

∣

∣

(

1

|Qjm|∫

Qjm

|aj,Gm (y)|rdy

)1/r

χjm(·)∣

∣

∣

∣

2)1/2

|Lpk

∥

∥

∥

∥

≤∥

∥

∥

∥

(

∑

j,G,m

(

M(|aj,G

m |r)(·))2/r)1/2

|Lpk

∥

∥

∥

∥

= ∥

∥

{

M(|aj,G

m |r)}|Lpk/r (�2/r )∥

∥

1/r �∥

∥

[

x �→ |aj,Gm (x)|r] | Lpk/r (�2/r )

∥

∥

1/r

= ∥

∥

[

x �→ aj,Gm (x)

] | Lpk(�2)

∥

∥,

where M denotes the Hardy-Littlewood maximal function and we have used theFefferman-Stein maximal inequality, by choosing 0 < r < min(p0,2).

If μ ∈ λq(φ), then

[

x �→ {

μj,Gm χjm(x)

}] ∈ �q(φ, �2) = (Lp0(�2),Lp1(�2))η,q .

Thus R[x �→ {μj,Gm χjm(x)}] ∈ (λp0 , λp1)η,q and

∥

∥R[

x �→ {

μj,Gm χjm(x)

}] | (λp0 , λp1)η,q

∥

∥ �∥

∥

[

x �→ {

μj,Gm χjm(x)

}] | �q(φ, �2)∥

∥,


that is,∥

∥

{∣

∣μj,Gm

∣

∣

} | (λp0 , λp1)η,q

∥

∥ �∥

∥

{

μj,Gm

} | λq(φ)∥

∥.

Applying now Lemma 6.3 to A0 = λp0 and A1 = λp1 , we get

‖μ | (λp0, λp1)η,q‖ � ‖|μ| | (λp0 , λp1)η,q‖,which completes the proof. �

Remark 6.5 It should be noted that the pair (λp0 , λp1) above is compatible forinterpolation. Indeed, one can show that both spaces are continuously embed-ded into the Hausdorff t.v.s. formed by all sequences μ ≡ {μj,G

m } ⊂ C such that[x �→ {μj,G

m χjm(x)}] ∈ Lp0(�2) + Lp1(�2).

Now we are ready to formulate the main statement on wavelet decompositions.

Theorem 6.6 Let 0 < q < ∞ and φ ∈ B with βφ > 0. Let also �k be the waveletsystem given by (6.3). Then there exists k(φ) ∈ N such that, for every N � k > k(φ)

there holds: f ∈ S ′ belongs to hq(φ) if, and only if, it can be represented as

f =∑

j,G,m

μj,Gm 2−jn/2ψ

j,Gm with μ = {μj,G

m } ∈ λq(φ), (6.8)

(summability in S ′). Moreover, the wavelet coefficients μj,Gm are uniquely determined

by

μj,Gm = μ

j,Gm (f ) := 2jn/2〈f,ψ

j,Gm 〉. (6.9)

Furthermore,

‖f | hq(φ)‖ ∼ ‖μ(f ) | λq(φ)‖ (equivalent quasi-norms), (6.10)

where μ(f ) ≡ {μj,Gm (f )}.

Proof Step 1: Suppose that f ∈S ′ can be represented as f =∑

j,G,m μj,Gm 2−jn/2ψ

j,Gm

(in S ′) for some μ = {μj,Gm } ∈ λq(φ). Choose p0,p1 such that 0 < 1

p1< βφ ≤ αφ <

1p0

< ∞. Taking a system �k with k large enough, then

T : λp0 + λp1 −→ hp0 + hp1 ,

given by

T μ :=∑

j,G,m

μj,Gm 2−jn/2ψ

j,Gm (in S ′),

is a well-defined linear operator and its restriction to each λpiis bounded into hpi

(cf.Proposition 6.1). By interpolation (using Corollary 4.5 and Lemma 6.4) we get

‖f | hq(φ)‖ = ‖T μ | hq(φ)‖ � ‖μ | λq(φ)‖.


Step 2: Let now f ∈ hq(φ). Taking p0 and p1 as in Step 1, Corollary 4.5 shows, inparticular, that f = f0 + f1 for some fi ∈ hpi

, i = 0,1. By Proposition 6.1 we have

fi = ∑

j,G,m

μj,Gm (fi)2−jn/2ψ

j,Gm (in S ′), where μ(fi) ∈ λpi

are given by μj,Gm (fi) =

2jn/2〈fi, ψj,Gm 〉 and the wavelet system �k is chosen again with large k. Hence we

have

f =∑

j,G,m

μj,Gm (f )2−jn/2ψ

j,Gm

in S ′, where we define μ(f ) := μ(f0) + μ(f1) (note that the quantity μ(f ) doesnot depend on the decomposition f0 + f1 = f chosen). Applying interpolation to theoperator

S : hp0 + hp1 −→ λp0 + λp1

defined by Sf := {μj,Gm (f )}, we conclude that μ(f ) ∈ λq(φ) and

‖μ(f ) | λq(φ)‖ = ‖Sf | λq(φ)‖ � ‖f | hq(φ)‖. (6.11)

From Step 1 we also see that

‖f | hq(φ)‖ � ‖μ(f ) | λq(φ)‖. (6.12)

Hence (6.10) will follow from (6.11) and (6.12) after we prove the uniqueness of therepresentation. We show this next:

Suppose that f ∈ hq(φ) is represented as f = ∑

j,G,m σj,Gm 2−jn/2ψ

j,Gm for

some σ ≡ {σ j,Gm } ∈ λq(φ). By Lemma 6.4 there are σ0 ∈ λp0 , σ1 ∈ λp1 such that

σ0 + σ1 = σ . Then fi := ∑

j,G,m σj,Gm,i 2−jn/2ψ

j,Gm belong to hpi

, i = 0,1. But thisdecomposition is unique, so that σi = μ(fi) with μ(fi) defined as above. Thereforeσ = μ(f0) + μ(f1) = μ(f ), since f = f0 + f1, which shows that the wavelet coef-ficients are, in fact, uniquely determined by (6.9). �

Remark 6.7 The symbol 〈f, ψj,Gm 〉 appearing in (6.9) is interpreted according to the

definition of the operator S in Step 2, as the sum of two quantities as described inRemark 6.2. More precisely,

〈f,ψj,Gm 〉 = 2jn/2(〈f0,ψ

j,Gm 〉 + 〈f1,ψ

j,Gm 〉),

where we can take any fi ∈ hpisuch that f0 + f1 = f .

Remark 6.8 As in Proposition 6.1, it is also possible to show that �k provides anunconditional basis in hq(φ). This will be proved elsewhere.

Remark 6.9 Using similar arguments it should also be possible to obtain subatomicdecompositions in the space hq(φ) (see [10] for further details).


Appendix

Proof of Lemma 4.1 It is not hard to see that (4.1) is a consequence of the estimate

K(t, T a;B0,B1) ≤ M0 K(M1 t/M0, a;A0,A1), t > 0. (7.1)

We show this inequality. For any t > 0 and a ∈ A0 + A1, we have

K(t, T a;B0,B1) = infb0+b1=T a

bi∈Bi

(‖b0|B0‖ + t‖b1|B1‖)

≤ infa0+a1=a

ai∈Ai

(‖b0|B0‖ + t‖b1|B1‖)

≤ infa0+a1=a

ai∈Ai

(

M0‖a0 | A0‖ + tM1‖a1 | A1‖)

= M0K(M1t/M0, a;A0,A1). �

Proof of Lemma 6.3 Let |f | ≤ |g|, with g ∈ (A0,A1)γ,q .Then, given any x ∈ I ,

√

(�f (x))2 + (�f (x))2 = |f (x)| ≤ |g(x)| = |a0(x) + a1(x)|,for some a0 ∈ Ak (independently of x), k = 0,1. Notice also that

|�f (x)|, |�f (x)| ≤ |a0(x)| + |a1(x)|,this being true for any decomposition of g = a0 + a1 with a0 ∈ A0, a1 ∈ A1.

Therefore there exists, for each x ∈ I , ξRk (x), ξ I

k (x) ∈ [0, |ak(x)|], k = 0,1, suchthat

|�f (x)| = ξR0 (x) + ξR

1 (x) and |�f (x)| = ξI0 (x) + ξI

1 (x).

Obviously, by the hypothesis (6.7), ξRk , ξ I

k ∈ Ak , k = 0,1, and

‖ξRk |Ak‖ � ‖ak|Ak‖, ‖ξI

k |Ak‖ � ‖ak|Ak‖ (k = 0,1),

so we have obtained, in particular, that

|�f |, |�f | ∈ A0 + A1.

Moreover, given any t > 0,

K(t, |�f |) = inf|�f |=b0+b1b0∈A0,b1∈A1

(‖b0|A0‖ + t ‖b1|A1‖)

≤ inf|�f |=ξ0+ξ1a0∈A0,a1∈A1with a0+a1=g

(‖ξR0 |A0‖ + t ‖ξR

1 |A1‖)


� infg=a0+a1

a0∈A0,a1∈A1

(‖a0|A0‖ + t ‖a1|A1‖) = K(t, g),

therefore |�f | ∈ (A0,A1)γ,q and

‖|�f ||(A0,A1)γ,q‖ � ‖g|(A0,A1)γ,q‖. (7.2)

Similarly, |�f | ∈ (A0,A1)γ,q and

‖|�f ||(A0,A1)γ,q‖ � ‖g|(A0,A1)γ,q‖. (7.3)

In order to extract now information both for �f and �f , let’s consider any realfunction h on I such that

|h| ∈ (A0,A1)γ,q and ‖|h| |(A0,A1)γ,q‖ � ‖g|(A0,A1)γ,q‖.Given any d0 ∈ A0, d1 ∈ A1 such that |h| = d0 + d1, we have, for any x ∈ I , that

either h(x) = d0(x) + d1(x) (in case h(x) ≥ 0) or h(x) = −d0(x) − d1(x) (in caseh(x) < 0). Let’s then define, for each x ∈ I ,

ζ0(x) ={

d0(x) if h(x) ≥ 0−d0(x) if h(x) < 0

and ζ1(x) ={

d1(x) if h(x) ≥ 0,−d1(x) if h(x) < 0,

so that h = ζ0 + ζ1, where, by the hypothesis (6.7), ζ0 ∈ A0, ζ1 ∈ A1 and

‖ζ0|A0‖ � ‖d0|A0‖ and ‖ζ1|A1‖ � ‖d1|A1‖.Therefore, h ∈ A0 + A1 and, moreover, proceeding similarly as before, K(t,h) �K(t, |h|) for any t > 0, hence

h ∈ (A0,A1)γ,q and ‖h|(A0,A1)γ,q‖ � ‖|h||(A0,A1)γ,q .

Applying this to the cases h = �f and h = �f , we get that �f,�f ∈ (A0,A1)γ,q

and, taking also (7.2) and (7.3) into account,

‖�f |(A0,A1)γ,q‖ � ‖g|(A0,A1)γ,q‖ and(7.4)

‖�f |(A0,A1)γ,q‖ � ‖g|(A0,A1)γ,q‖.Consequently, f = �f + i�f also belongs to (A0,A1)γ,q . Furthermore, since,

given any decompositions �f = aR0 + aR

1 , �f = aI0 + aI

1 , with aR0 , aI

0 ∈ A0, aR1 ,

aI1 ∈ A1, we have

f = aR0 + aR

1 + iaI0 + iaI

1 = aR0 + iaI

0 + aR1 + iaI

1 ,

with aR0 + iaI

0 ∈ A0 and aR1 + iaI

1 ∈ A1, then, given any t > 0,

K(t, f ) � inf�f =aR

0 +aR1 ,�f =aI

0 +aI1

aR0 ,aI

0 ∈A0;aR1 ,aI

1 ∈A1

(‖aR0 |A0‖ + ‖aI

0 |A0‖ + t ‖aR1 |A1‖ + t ‖aI

1 |A1‖)

= K(t,�f ) + K(t,�f ),


hence, finally, with the help of (7.4),

‖f |(A0,A1)γ,q‖ � ‖�f |(A0,A1)γ,q‖ + ‖�f |(A0,A1)γ,q‖ � ‖g|(A0,A1)γ,q‖.�

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real interpolation of generalized besov-hardy spaces and applications

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