algebras of singular integral operators on nilpotent...
TRANSCRIPT
![Page 1: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/1.jpg)
Algebras of singular integral operatorson nilpotent groups
(joint work with A. Nagel, E.M. Stein, S. Wainger)
Fulvio Ricci
Scuola Normale Superiore, Pisa
Interactions of Harmonic Analysis and Operator TheoryUniversity of BirminghamSeptember 13-16, 2016
![Page 2: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/2.jpg)
Dilations and CZ theory on nilpotent groups
In the classical extension of Calderón-Zygmund theory to nilpotent groups
(Folland-Stein, Korányi) it is important to assume that the scale-invariance
properties of the singular kernels be adapted to dilations which are group
automorphisms.
This is a strong constraint. For instance, the isotropic dilations on the
underlying vector space structure on the Lie algebra are not allowed as soon
as the group is not abelian.
However, several situations have been encountered in the past where
non-automorphic dilations must be considered, at least locally.
The typical case is that of the isotropic dilations on the Heisenberg group:
Phong-Stein, 1982
Müller, R., Stein, 1995
Nagel, Stein, 2006
Müller, Peloso, R., 2015
![Page 3: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/3.jpg)
Dilations and CZ theory on nilpotent groups
In the classical extension of Calderón-Zygmund theory to nilpotent groups
(Folland-Stein, Korányi) it is important to assume that the scale-invariance
properties of the singular kernels be adapted to dilations which are group
automorphisms.
This is a strong constraint. For instance, the isotropic dilations on the
underlying vector space structure on the Lie algebra are not allowed as soon
as the group is not abelian.
However, several situations have been encountered in the past where
non-automorphic dilations must be considered, at least locally.
The typical case is that of the isotropic dilations on the Heisenberg group:
Phong-Stein, 1982
Müller, R., Stein, 1995
Nagel, Stein, 2006
Müller, Peloso, R., 2015
![Page 4: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/4.jpg)
Dilations and CZ theory on nilpotent groups
In the classical extension of Calderón-Zygmund theory to nilpotent groups
(Folland-Stein, Korányi) it is important to assume that the scale-invariance
properties of the singular kernels be adapted to dilations which are group
automorphisms.
This is a strong constraint. For instance, the isotropic dilations on the
underlying vector space structure on the Lie algebra are not allowed as soon
as the group is not abelian.
However, several situations have been encountered in the past where
non-automorphic dilations must be considered, at least locally.
The typical case is that of the isotropic dilations on the Heisenberg group:
Phong-Stein, 1982
Müller, R., Stein, 1995
Nagel, Stein, 2006
Müller, Peloso, R., 2015
![Page 5: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/5.jpg)
Product theory
More precisely, the common theme in the above mentioned papers is the
simultaneous presence of isotropic dilations and the standard automorphic
(parabolic) dilations and their combination in a (non-automorphic)
two-parameter dilation structure calling for some adaptation of the product
theory of singular integrals on Rn.
The limitations imposed by the compatibility with the group structure have
produced a restricted form of product theory in which the two dilation
parameters are subject to a one-sided limitation (flag kernels, Nagel, R.,
Stein, 2001, Nagel, R., Stein, Wainger, 2012).
![Page 6: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/6.jpg)
Product theory
More precisely, the common theme in the above mentioned papers is the
simultaneous presence of isotropic dilations and the standard automorphic
(parabolic) dilations and their combination in a (non-automorphic)
two-parameter dilation structure calling for some adaptation of the product
theory of singular integrals on Rn.
The limitations imposed by the compatibility with the group structure have
produced a restricted form of product theory in which the two dilation
parameters are subject to a one-sided limitation (flag kernels, Nagel, R.,
Stein, 2001, Nagel, R., Stein, Wainger, 2012).
![Page 7: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/7.jpg)
General dilations on Rn
For a = (a1, a2, . . . , an) ∈ Rn+ define the a-dilations
x = (x1, x2, . . . , xn) 7−→ (t1/a1 x1, t1/a2 x2, . . . , t1/an xn) = δa(t)x
Homogenous norm:
|x |a = |x1|a1 + |x2|a2 + · · ·+ |xn|an .
Homogeneous dimension:
Qa =1a1
+1a2
+ ·+ 1an
Let k = (k1, k2, . . . , kn) ∈ Nn be a multiindex. We set
∂kx = ∂
k1x1∂
k2x2· · · ∂kn
xn .
![Page 8: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/8.jpg)
Calderón-Zygmund kernels adapted to the a-dilations
DefinitionA smooth CZ kernel of type a is a distribution K which is C∞ away from 0
and satisfies
• differential inequalities: for any k and any x 6= 0,∣∣∂kx K (x)
∣∣ ≤ Ck|x |−Qa−[k]aa ,
with [k]a =∑
kj/aj ;
• cancellations: ∣∣∣ ∫ K (x)ϕ(δa(t)x
)dx∣∣∣ ≤ C‖ϕ‖C1 ,
for all ϕ ∈ C1c (B), B a fixed “unit ball”, and every t > 0.
![Page 9: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/9.jpg)
Calderón-Zygmund kernels adapted to the a-dilations
DefinitionA smooth CZ kernel of type a is a distribution K which is C∞ away from 0
and satisfies
• differential inequalities: for any k and any x 6= 0,∣∣∂kx K (x)
∣∣ ≤ Ck|x |−Qa−[k]aa ,
with [k]a =∑
kj/aj ;
• cancellations: ∣∣∣ ∫ K (x)ϕ(δa(t)x
)dx∣∣∣ ≤ C‖ϕ‖C1 ,
for all ϕ ∈ C1c (B), B a fixed “unit ball”, and every t > 0.
![Page 10: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/10.jpg)
Calderón-Zygmund kernels adapted to the a-dilations
DefinitionA smooth CZ kernel of type a is a distribution K which is C∞ away from 0
and satisfies
• differential inequalities: for any k and any x 6= 0,∣∣∂kx K (x)
∣∣ ≤ Ck|x |−Qa−[k]aa ,
with [k]a =∑
kj/aj ;
• cancellations: ∣∣∣ ∫ K (x)ϕ(δa(t)x
)dx∣∣∣ ≤ C‖ϕ‖C1 ,
for all ϕ ∈ C1c (B), B a fixed “unit ball”, and every t > 0.
![Page 11: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/11.jpg)
Equivalent characterizations
TheoremFor a distribution K ∈ S ′ the following are equivalent:
(i) K is a CZ kernel of type a;
(ii) the Fourier transform K = m is a bounded function, smooth away from
the origin and satisfying∣∣∂kξm(ξ) ≤ Ck|ξ|
−∑
kj/aja ;
(iii)
K =∑
j∈Z2Qa jϕj
(δa(2j )x
)=∑
j∈Zϕ
(j)j (x) ,
where the ϕj are C∞ functions supported on the unit ball B, bounded in
any Cm-norm and with ∫ϕj (x) dx = 0 .
![Page 12: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/12.jpg)
Equivalent characterizations
TheoremFor a distribution K ∈ S ′ the following are equivalent:
(i) K is a CZ kernel of type a;
(ii) the Fourier transform K = m is a bounded function, smooth away from
the origin and satisfying∣∣∂kξm(ξ) ≤ Ck|ξ|
−∑
kj/aja ;
(iii)
K =∑
j∈Z2Qa jϕj
(δa(2j )x
)=∑
j∈Zϕ
(j)j (x) ,
where the ϕj are C∞ functions supported on the unit ball B, bounded in
any Cm-norm and with ∫ϕj (x) dx = 0 .
![Page 13: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/13.jpg)
Equivalent characterizations
TheoremFor a distribution K ∈ S ′ the following are equivalent:
(i) K is a CZ kernel of type a;
(ii) the Fourier transform K = m is a bounded function, smooth away from
the origin and satisfying∣∣∂kξm(ξ) ≤ Ck|ξ|
−∑
kj/aja ;
(iii)
K =∑
j∈Z2Qa jϕj
(δa(2j )x
)=∑
j∈Zϕ
(j)j (x) ,
where the ϕj are C∞ functions supported on the unit ball B, bounded in
any Cm-norm and with ∫ϕj (x) dx = 0 .
![Page 14: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/14.jpg)
Equivalent characterizations
TheoremFor a distribution K ∈ S ′ the following are equivalent:
(i) K is a CZ kernel of type a;
(ii) the Fourier transform K = m is a bounded function, smooth away from
the origin and satisfying∣∣∂kξm(ξ) ≤ Ck|ξ|
−∑
kj/aja ;
(iii)
K =∑
j∈Z2Qa jϕj
(δa(2j )x
)=∑
j∈Zϕ
(j)j (x) ,
where the ϕj are C∞ functions supported on the unit ball B, bounded in
any Cm-norm and with ∫ϕj (x) dx = 0 .
![Page 15: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/15.jpg)
Proper kernels
In the rest of this talk we want to restrict our attention to “proper” kernels,
which exhibit a CZ singularity at the origin, but combined with a Schwartz
decay at infinity.
This involves modifying the previous conditions in the following way:
• differential inequalities of the kernel K :∣∣∂kx K (x)
∣∣ ≤ Ck,N |x |−
∑(1+kj )/aj
a
• differential inequalities of the multiplier m:∣∣∂kξm(ξ)
∣∣ ≤ Ck|ξ|−
∑kj/aj
a
• dyadic decomposition:
K =∑j∈Z
ϕ(j)j
![Page 16: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/16.jpg)
Proper kernels
In the rest of this talk we want to restrict our attention to “proper” kernels,
which keep the CZ singularity at the origin and are infinitely regular at infinity,
i.e. with a Schwartz decay.
This involves modifying the previous conditions in the following way:
• differential inequalities of the kernel K :∣∣∂kx K (x)
∣∣ ≤ Ck,N |x |−
∑(1+kj )/aj
a(1 + |x |
)−N
• differential inequalities of the multiplier m:∣∣∂kξm(ξ)
∣∣ ≤ Ck|ξ|−
∑kj/aj
a
• dyadic decomposition:
K =∑j∈Z
ϕ(j)j
![Page 17: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/17.jpg)
Proper kernels
In the rest of this talk we want to restrict our attention to “proper” kernels,
which keep the CZ singularity at the origin and are infinitely regular at infinity,
i.e. with a Schwartz decay.
This involves modifying the previous conditions in the following way:
• differential inequalities of the kernel K :∣∣∂kx K (x)
∣∣ ≤ Ck,N |x |−
∑(1+kj )/aj
a(1 + |x |
)−N
• differential inequalities of the multiplier m:∣∣∂kξm(ξ)
∣∣ ≤ Ck(1 + |ξ|a
)−∑kj/aj ;
• dyadic decomposition:
K =∑j∈Z
ϕ(j)j
![Page 18: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/18.jpg)
Proper kernels
In the rest of this talk we want to restrict our attention to “proper” kernels,
which keep the CZ singularity at the origin and are infinitely regular at infinity,
i.e. with a Schwartz decay.
This involves modifying the previous conditions in the following way:
• differential inequalities of the kernel K :∣∣∂kx K (x)
∣∣ ≤ Ck,N |x |−
∑(1+kj )/aj
a(1 + |x |
)−N
• differential inequalities of the multiplier m:∣∣∂kξm(ξ)
∣∣ ≤ Ck(1 + |ξ|a
)−∑kj/aj ;
• dyadic decomposition:
K = η +∑j≥0
ϕ(j)j , η ∈ S
![Page 19: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/19.jpg)
Composition of CZ kernels with different homogeneities
Consider now two proper CZ kernels, K1,K2, adapted to dilations of type a
and b respectively. We want to understand what kind of estimates are
satisfied by the convolution K1 ∗ K2.
It is quite obvious that the convolution will be C∞ away from the origin
(pseudolocality), with Schwartz decay at infinity. A more refined question is
what differential inequalities it satisfies near the origin.
We take a model example in two variables.
On R2 we denote by z = (x , y) the space variables and ζ = (ξ, η) the
frequency variables.
a = (1, 1) (isotropic dilations), with |z|a = |x |+ |y |, Qa = 2,
b = (2, 1) (parabolic dilations), with |z|b = |x |2 + |y |, Qb = 3/2.
![Page 20: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/20.jpg)
Composition of CZ kernels with different homogeneities
Consider now two proper CZ kernels, K1,K2, adapted to dilations of type a
and b respectively. We want to understand what kind of estimates are
satisfied by the convolution K1 ∗ K2.
It is quite obvious that the convolution will be C∞ away from the origin
(pseudolocality), with Schwartz decay at infinity. A more refined question is
what differential inequalities it satisfies near the origin.
We take a model example in two variables.
On R2 we denote by z = (x , y) the space variables and ζ = (ξ, η) the
frequency variables.
a = (1, 1) (isotropic dilations), with |z|a = |x |+ |y |, Qa = 2,
b = (2, 1) (parabolic dilations), with |z|b = |x |2 + |y |, Qb = 3/2.
![Page 21: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/21.jpg)
The multiplier side
It suffices to consider points ζ = (ξ, η) with |ξ|, |η| > 1.
Set m1 = K1, m2 = K2, Then∣∣∂ jm1(ζ)∣∣ . (1 + |ξ|+ |η|
)−j1−j2∣∣∂ lm2(ζ)∣∣ . (1 + |ξ|2 + |η|
)− 12 l1−l2 .
We estimate the derivatives of m = K1 ∗ K2 = m1m2.
![Page 22: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/22.jpg)
The multiplier side
It suffices to consider points ζ = (ξ, η) with |ξ|, |η| > 1.
Set m1 = K1, m2 = K2, Then∣∣∂ jm1(ζ)∣∣ . (1 + |ξ|+ |η|
)−j1−j2∣∣∂ lm2(ζ)∣∣ . (1 + |ξ|2 + |η|
)− 12 l1−l2 .
We estimate the derivatives of m = K1 ∗ K2 = m1m2.
![Page 23: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/23.jpg)
Three regions
(I) For |η| < |ξ|,∣∣∂km(ζ)
∣∣ . |ξ|−k1−k2
(II) For |η|12 < |ξ| < |η|,
∣∣∂km(ζ)∣∣ . |ξ|−k1 |η|−k2
(III) For |ξ| < |η|12 ,
∣∣∂km(ζ)∣∣ . |η|− k1
2 −k2
These inequalities are subsumed by the single formula∣∣∂km(ζ)∣∣ . (1 + |ξ|+ |η|
12)−k1
(1 + |ξ|+ |η|
)−k2
∼=(1 + |ζ|
12b
)−k1(1 + |ζ|a
)−k2 .
![Page 24: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/24.jpg)
Three regions
(I) For |η| < |ξ|,∣∣∂km(ζ)
∣∣ . |ξ|−k1−k2
(II) For |η|12 < |ξ| < |η|,
∣∣∂km(ζ)∣∣ . |ξ|−k1 |η|−k2
(III) For |ξ| < |η|12 ,
∣∣∂km(ζ)∣∣ . |η|− k1
2 −k2
These inequalities are subsumed by the single formula∣∣∂km(ζ)∣∣ . (1 + |ξ|+ |η|
12)−k1
(1 + |ξ|+ |η|
)−k2
∼=(1 + |ζ|
12b
)−k1(1 + |ζ|a
)−k2 .
![Page 25: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/25.jpg)
Three regions
(I) For |η| < |ξ|,∣∣∂km(ζ)
∣∣ . |ξ|−k1−k2
(II) For |η|12 < |ξ| < |η|,
∣∣∂km(ζ)∣∣ . |ξ|−k1 |η|−k2
(III) For |ξ| < |η|12 ,
∣∣∂km(ζ)∣∣ . |η|− k1
2 −k2
These inequalities are subsumed by the single formula∣∣∂km(ζ)∣∣ . (1 + |ξ|+ |η|
12)−k1
(1 + |ξ|+ |η|
)−k2
∼=(1 + |ζ|
12b
)−k1(1 + |ζ|a
)−k2 .
![Page 26: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/26.jpg)
Three regions
(I) For |η| < |ξ|,∣∣∂km(ζ)
∣∣ . |ξ|−k1−k2
(II) For |η|12 < |ξ| < |η|,
∣∣∂km(ζ)∣∣ . |ξ|−k1 |η|−k2
(III) For |ξ| < |η|12 ,
∣∣∂km(ζ)∣∣ . |η|− k1
2 −k2
These inequalities are subsumed by the single formula∣∣∂km(ζ)∣∣ . (1 + |ξ|+ |η|
12)−k1
(1 + |ξ|+ |η|
)−k2
∼=(1 + |ζ|
12b
)−k1(1 + |ζ|a
)−k2 .
![Page 27: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/27.jpg)
Three regions
(I) For |η| < |ξ|,∣∣∂km(ζ)
∣∣ . |ξ|−k1−k2
(II) For |η|12 < |ξ| < |η|,
∣∣∂km(ζ)∣∣ . |ξ|−k1 |η|−k2
(III) For |ξ| < |η|12 ,
∣∣∂km(ζ)∣∣ . |η|− k1
2 −k2
These inequalities are subsumed by the single formula∣∣∂km(ζ)∣∣ . (1 + |ξ|+ |η|
12)−k1
(1 + |ξ|+ |η|
)−k2
∼=(1 + |ζ|
12b
)−k1(1 + |ζ|a
)−k2 .
![Page 28: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/28.jpg)
-
6
1
1
|ξ|
|η|
������������������
.
..............................
..................................
.....................................
.........................................
............................................
................................................
...................................................
.......................................................
..........................................................
..............................................................
.................................................................
....................................................................
(I)
isotropic
(II)
product
(III)
parabolic
![Page 29: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/29.jpg)
-
6
1
1
|ξ|
|η|
������������������
.
..............................
..................................
.....................................
.........................................
............................................
................................................
...................................................
.......................................................
..........................................................
..............................................................
.................................................................
....................................................................
2j 2j+12j−1
22(j+1)
22j
22(j−1)
2j2j+1
...
![Page 30: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/30.jpg)
Dyadic scales
The scales of the rectangles are the following:
(I) (2j , 2j ) in the isotropic region,
(II) (2j , 2k ) with j ≤ k ≤ 2j in the product region,
(III) (2j , 22j ) in the parabolic region.
The rectangles in (II) do not intersect the axes.
Those in (I) and (III) intersect one of the two axes, but do not contain the
origin.
![Page 31: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/31.jpg)
Dyadic scales
The scales of the rectangles are the following:
(I) (2j , 2j ) in the isotropic region,
(II) (2j , 2k ) with j ≤ k ≤ 2j in the product region,
(III) (2j , 22j ) in the parabolic region.
The rectangles in (II) do not intersect the axes.
Those in (I) and (III) intersect one of the two axes, but do not contain the
origin.
![Page 32: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/32.jpg)
Dyadic scales
The scales of the rectangles are the following:
(I) (2j , 2j ) in the isotropic region,
(II) (2j , 2k ) with j ≤ k ≤ 2j in the product region,
(III) (2j , 22j ) in the parabolic region.
The rectangles in (II) do not intersect the axes.
Those in (I) and (III) intersect one of the two axes, but do not contain the
origin.
![Page 33: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/33.jpg)
Dyadic scales
The scales of the rectangles are the following:
(I) (2j , 2j ) in the isotropic region,
(II) (2j , 2k ) with j ≤ k ≤ 2j in the product region,
(III) (2j , 22j ) in the parabolic region.
The rectangles in (II) do not intersect the axes.
Those in (I) and (III) intersect one of the two axes, but do not contain the
origin.
![Page 34: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/34.jpg)
Dyadic scales
The scales of the rectangles are the following:
(I) (2j , 2j ) in the isotropic region,
(II) (2j , 2k ) with j ≤ k ≤ 2j in the product region,
(III) (2j , 22j ) in the parabolic region.
The rectangles in (II) do not intersect the axes.
Those in (I) and (III) intersect one of the two axes, but do not contain the
origin.
![Page 35: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/35.jpg)
-
6
j
k
��������������������
����������������������
k = j
k = 2j
•
• •
◦
•
•
◦
◦
•
•
◦
◦
◦
•
•
◦
◦
◦
•
◦
◦
•
◦ •
![Page 36: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/36.jpg)
Dyadic decomposition of the multiplier
The multiplier m can be decomposed
m = m0 + m(I) + m(II) + m(III)
= m0 +∑j≥0
m(I)j (2−jξ, 2−jη)
+∑
j≤k≤2j
m(II)jk (2−jξ, 2−kη)
+∑j≥0
m(III)j (2−jξ, 2−2jη) ,
where all the summands are smooth functions supported in the unit ball B,
uniformly bounded in all Ck -norms and
• the terms m(I)j ,m
(III)j vanish at 0,
• the m(II)jk vanish on the coordinate axes.
![Page 37: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/37.jpg)
Dyadic decomposition of the multiplier
The multiplier m can be decomposed
m = m0 + m(I) + m(II) + m(III)
= m0 +∑j≥0
m(I)j (2−jξ, 2−jη)
+∑
j≤k≤2j
m(II)jk (2−jξ, 2−kη)
+∑j≥0
m(III)j (2−jξ, 2−2jη) ,
where all the summands are smooth functions supported in the unit ball B,
uniformly bounded in all Ck -norms and
• the terms m(I)j ,m
(III)j vanish at 0,
• the m(II)jk vanish on the coordinate axes.
![Page 38: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/38.jpg)
The kernel in dyadic form
On the other side of the Fourier transform,
K1 ∗ K2 = η + K (I) + K (II) + K (III) ,
where η ∈ S, K (I),K (III) are proper CZ kernels, the first isotropic and the
second parabolic, while
K (II) =∑
j≤k≤2j
2j+kϕjk (2jx , 2k y) . (1)
The ϕjk are uniformly bounded in every Schwartz norm and∫ϕjk (x , y) dx = 0 ∀ y ,
∫ϕjk (x , y) dy = 0 ∀ x .
As a further refinement, each ϕjk can be replaced by a dyadic sum, with
rapidly decreasing coefficients, of smooth functions with compact support
and the same cancellations.
In this way the summands in (1) can be assumed to be supported in B and
uniformly bounded in each Ck -norm.
![Page 39: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/39.jpg)
The kernel in dyadic form
On the other side of the Fourier transform,
K1 ∗ K2 = η + K (I) + K (II) + K (III) ,
where η ∈ S, K (I),K (III) are proper CZ kernels, the first isotropic and the
second parabolic, while
K (II) =∑
j≤k≤2j
2j+kϕjk (2jx , 2k y) . (1)
The ϕjk are uniformly bounded in every Schwartz norm and∫ϕjk (x , y) dx = 0 ∀ y ,
∫ϕjk (x , y) dy = 0 ∀ x .
As a further refinement, each ϕjk can be replaced by a dyadic sum, with
rapidly decreasing coefficients, of smooth functions with compact support
and the same cancellations.
In this way the summands in (1) can be assumed to be supported in B and
uniformly bounded in each Ck -norm.
![Page 40: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/40.jpg)
Outside of CZ theory
The resulting kernel K = K (I) + K (II) + K (III) is not a CZ kernel. It satisfies the
differential inequalities∣∣∂px ∂
qy K (z)
∣∣ ≤ CpqN |z|−1−pa |z|−1−q
b
(1 + |z|
)−N (2)
for all p, q,N, plus cancellations in each variable separately.
Condition (2) is weaker than any type of CZ condition. In fact it gives the
estimate ∣∣{z : |K (z)| > α}∣∣ . logα
α,
and no better in general.
![Page 41: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/41.jpg)
New classes of kernels
We consider n different homogeneities on Rn, with homogeneous norms
|x |e1 = |x1|e(1,1) + |x2|e(1,2) + · · ·+ |xn|e(1,n)
· · · · · ·
|x |en = |x1|e(n,1) + |x2|e(n,2) + · · ·+ |xn|e(n,n) .
It will be necessary to also consider the situation where each xj is itself a
multivariate component in Rdj , in a higher dimensional space
Rd = Rd1+d2+···+dn . In this case |xj | denotes a homogeneous norm adapted to
given dilations on Rdj .
![Page 42: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/42.jpg)
Multi-norm inequalities
We want to consider kernels K which satisfy the following inequalities:
(*)∣∣∂k
x K (x)∣∣ ≤ CkN |x |−1−k1
e1|x |−1−k2
e2· · · |x |−1−kn
en
(1+|x |
)−N.
It is natural to impose the following basic assumptions:
e(j, j) = 1 , e(j, k) ≤ e(j, l)e(l, k)
for all j, k , l .
Under the basic assumptions, the inequalities (*) hold, e.g., for proper CZ
kernels of type a = e1, e2, . . . , en.
![Page 43: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/43.jpg)
Multi-norm inequalities
We want to consider kernels K which satisfy the following inequalities:
(*)∣∣∂k
x K (x)∣∣ ≤ CkN |x |−1−k1
e1|x |−1−k2
e2· · · |x |−1−kn
en
(1+|x |
)−N.
It is natural to impose the following basic assumptions:
e(j, j) = 1 , e(j, k) ≤ e(j, l)e(l, k)
for all j, k , l .
Under the basic assumptions, the inequalities (*) hold, e.g., for proper CZ
kernels of type a = e1, e2, . . . , en.
![Page 44: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/44.jpg)
The classes P0(E)
Denote by E the matrix(e(j, k)
)jk . We will always assume that the basic
assumptions are satisfied.
DefinitionThe class P0(E) consists of the distributions K which are smooth away from
the origin, satisfy the differential inequalities
(*)∣∣∂k
x K (x)∣∣ ≤ CkN |x |−1−k1
e1|x |−1−k2
e2· · · |x |−1−kn
en
(1+|x |
)−N,
for all k,N, plus appropriate cancellations in each variable xj .
Roughly speaking, the cancellations are defined inductively by stating that,
once we integrate K against a scaled bump function in a single variable xj ,
we obtain a kernel of the same kind in the remaining variables.
We will give two characterizations of the kernels in P0(E), one in terms of
their Fourier transforms, the other in terms of dyadic decompositions.
![Page 45: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/45.jpg)
The classes P0(E)
Denote by E the matrix(e(j, k)
)jk . We will always assume that the basic
assumptions are satisfied.
DefinitionThe class P0(E) consists of the distributions K which are smooth away from
the origin, satisfy the differential inequalities
(*)∣∣∂k
x K (x)∣∣ ≤ CkN |x |−1−k1
e1|x |−1−k2
e2· · · |x |−1−kn
en
(1+|x |
)−N,
for all k,N, plus appropriate cancellations in each variable xj .
Roughly speaking, the cancellations are defined inductively by stating that,
once we integrate K against a scaled bump function in a single variable xj ,
we obtain a kernel of the same kind in the remaining variables.
We will give two characterizations of the kernels in P0(E), one in terms of
their Fourier transforms, the other in terms of dyadic decompositions.
![Page 46: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/46.jpg)
The classes P0(E)
Denote by E the matrix(e(j, k)
)jk . We will always assume that the basic
assumptions are satisfied.
DefinitionThe class P0(E) consists of the distributions K which are smooth away from
the origin, satisfy the differential inequalities
(*)∣∣∂k
x K (x)∣∣ ≤ CkN |x |−1−k1
e1|x |−1−k2
e2· · · |x |−1−kn
en
(1+|x |
)−N,
for all k,N, plus appropriate cancellations in each variable xj .
Roughly speaking, the cancellations are defined inductively by stating that,
once we integrate K against a scaled bump function in a single variable xj ,
we obtain a kernel of the same kind in the remaining variables.
We will give two characterizations of the kernels in P0(E), one in terms of
their Fourier transforms, the other in terms of dyadic decompositions.
![Page 47: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/47.jpg)
Characterization via multipliers
In the frequency space we introduce a new family of norms:
|ξ|e1= |ξ1|1/e(1,1) + |ξ2|1/e(2,1) + · · ·+ |ξn|1/e(n,1)
· · · · · ·
|ξ|en = |ξ1|1/e(1,n) + |ξ2|1/e(2,n) + · · ·+ |ξn|1/e(n,n) .
LemmaA distribution K is in P0(E) if and only if m = K is a smooth function
satisfying the inequalities∣∣∂kξm(ξ)
∣∣ ≤ CkN(1 + |ξ|e1
)−k1 · · ·(1 + |ξ|en
)−kn .
![Page 48: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/48.jpg)
The cone Γ(E)
Denote by Γ(E) the cone in the positive orthant Rn+ defined by the inequalities
1e(l, k)
tl ≤ tk ≤ e(k , l)tl ∀ k , l
Let
m(ξ) =∑
J∈Γ(E)∩Nn
mJ (2−j1ξ1, . . . , 2−jnξn) ,
where the functions mprJ are supported in B, vanish on a neighborhood of the
coordinate hyperplanes and are uniformly bounded in every Ck -norm.
Then m satisfies the above inequalities, hence F−1m ∈ P0(E).
Up to “boundary terms”, it is also true that every m ∈ FP0(E) can be
decomposed as a dyadic sum of this kind. We will explain where the
“boundary terms” come from.
![Page 49: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/49.jpg)
The cone Γ(E)
Denote by Γ(E) the cone in the positive orthant Rn+ defined by the inequalities
1e(l, k)
tl ≤ tk ≤ e(k , l)tl ∀ k , l
Let
m(ξ) =∑
J∈Γ(E)∩Nn
mJ (2−j1ξ1, . . . , 2−jnξn) ,
where the functions mprJ are supported in B, vanish on a neighborhood of the
coordinate hyperplanes and are uniformly bounded in every Ck -norm.
Then m satisfies the above inequalities, hence F−1m ∈ P0(E).
Up to “boundary terms”, it is also true that every m ∈ FP0(E) can be
decomposed as a dyadic sum of this kind. We will explain where the
“boundary terms” come from.
![Page 50: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/50.jpg)
Dominant variables
We decompose the complement of B in the ξ-space into regions,
considering, for each j , which variable is dominant in the norm |ξ|ek.
We say that ξj is dominant in |ξ|ekat a point ξ if
|ξj |1/e(j,k) > |ξl |1/e(l,k) ∀l 6= j
After removing a finite union of hypersurfaces, at each point there is a unique
dominant variable in each norm (regular points).
LemmaIf ξk is dominant in |ξ|ej
at a regular point ξ, with j 6= k, then ξk is also
dominant in |ξ|ekat ξ.
![Page 51: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/51.jpg)
Marked partitions
To each regular point ξ we can associate a marked partition, i.e., a partition
{I1, . . . , Is} of {1, 2, . . . , n}, together with a distinguished element kr ∈ Ir for
each r = 1, . . . , s.
For instance, if ξ ∈ R5 is associated to the marked partition
{1, 3}, {2, 4, 5}
this means that ξ3 is dominat in |ξ|e1and in |ξ|e3
, while ξ4 is dominat in |ξ|e2,
|ξ|e4and |ξ|e5
.
To each marked partition S = {I1, . . . , Is; k1, . . . , ks} we associate the
(possibly empty) set ES of those regular ξ such that, for all r , ξkr is dominant
in |ξ|ejfor all j ∈ Ir .
Then
m = m0 +∑
S
mS ,
where m0 is supported on B and each mS near ES .
![Page 52: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/52.jpg)
Marked partitions
To each regular point ξ we can associate a marked partition, i.e., a partition
{I1, . . . , Is} of {1, 2, . . . , n}, together with a distinguished element kr ∈ Ir for
each r = 1, . . . , s.
For instance, if ξ ∈ R5 is associated to the marked partition
{1, 3}, {2, 4, 5}
this means that ξ3 is dominat in |ξ|e1and in |ξ|e3
, while ξ4 is dominat in |ξ|e2,
|ξ|e4and |ξ|e5
.
To each marked partition S = {I1, . . . , Is; k1, . . . , ks} we associate the
(possibly empty) set ES of those regular ξ such that, for all r , ξkr is dominant
in |ξ|ejfor all j ∈ Ir .
Then
m = m0 +∑
S
mS ,
where m0 is supported on B and each mS near ES .
![Page 53: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/53.jpg)
������������������������������������
TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT
�����������
�����
�����
@@
@@
@@@
@@@
@@@
@@
@@@
@@
@@
@@@
@@
������
���
������
�����������������
����
���� B
BBBBBBBBBBBBBB
BBBBBBB
BBBBBBB
HHHH
HHHH
HHHH
HHHH
HHH
HHHH
HHHH
HHHH
H
HHHH
HHHH
HHHH
H�����Q
QQQQ
CCCCCCCC
v3
v1
v2
v3
v1
v2
1
e(3, 1)
e(1, 2)
e(3, 2)
e(1, 3)
1
e(1, 2)
e(2, 3)
e(1, 3)e(2, 1)
1
e(2, 3)
e(3, 1)
e(2, 1)
e(3, 2)
{·1}{
·2}{
·3}
{·1, 2, 3}
{1,·2, 3}
{1, 2,·3}
{·1, 2}{
·3}
{·1}{2,
·3}
{1,·3}{
·2}
{1,·2}{
·3}
{·1}{·2, 3}
{·1, 3}{
·2}
1
![Page 54: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/54.jpg)
Dyadic decomposition of K
Putting all together,
m(ξ) = m0(ξ) +∑
S
∑J∈Γ(ES )∩Ns
mSJ (2−j1 · ξI1 , . . . , 2
−js · ξIs )
where, for each S, the functions mSJ are supported in B, vanish on a
neighborhood of the subspaces {ξ : ξI1 = 0}, . . . {ξ : ξIs = 0} and are
uniformly bounded in every Ck -norm.
Undoing the Fourier transform,
K (x) = η(x) +∑
S
∑J∈Γ(ES )∩Ns
2Q1 j1+···+Qs jsϕSJ (2j1 · xI1 , . . . , 2
js · xIs )
where η ∈ S and, for each S, the functions ϕSJ are uniformly bounded in every
Schwartz norm and satisfy the cancellations∫ϕS
J (xI1 , . . . , xIr , . . . , xIs ) dxIr = 0 ∀ xI1 , . . . , xIr−1 , xIr+1 , . . . , xIs
![Page 55: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/55.jpg)
Dyadic decomposition of K
Putting all together,
m(ξ) = m0(ξ) +∑
S
∑J∈Γ(ES )∩Ns
mSJ (2−j1 · ξI1 , . . . , 2
−js · ξIs )
where, for each S, the functions mSJ are supported in B, vanish on a
neighborhood of the subspaces {ξ : ξI1 = 0}, . . . {ξ : ξIs = 0} and are
uniformly bounded in every Ck -norm.
Undoing the Fourier transform,
K (x) = η(x) +∑
S
∑J∈Γ(ES )∩Ns
2Q1 j1+···+Qs jsϕSJ (2j1 · xI1 , . . . , 2
js · xIs )
where η ∈ S and, for each S, the functions ϕSJ are uniformly bounded in every
Schwartz norm and satisfy the cancellations∫ϕS
J (xI1 , . . . , xIr , . . . , xIs ) dxIr = 0 ∀ xI1 , . . . , xIr−1 , xIr+1 , . . . , xIs
![Page 56: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/56.jpg)
Equivalent characterizations of P0(E)
TheoremLet E be a matrix satisfying the basic assumptions. The following are
equivalent for K ∈ S ′(Rn):
• K ∈ P0(E),
• m = K is a smooth function satisfying the inequalities∣∣∂kξm(ξ)
∣∣ ≤ CkN(1 + |ξ|e1
)−k1 · · ·(1 + |ξ|en
)−kn ,
• K (x) = η(x) +∑
S
∑J∈Γ(ES )∩Ns 2Q1 j1+···+Qs jsϕS
J (2j1 · xI1 , . . . , 2js · xIs )
as above, or even with the additional condition suppϕSJ ⊂ B.
![Page 57: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/57.jpg)
Equivalent characterizations of P0(E)
TheoremLet E be a matrix satisfying the basic assumptions. The following are
equivalent for K ∈ S ′(Rn):
• K ∈ P0(E),
• m = K is a smooth function satisfying the inequalities∣∣∂kξm(ξ)
∣∣ ≤ CkN(1 + |ξ|e1
)−k1 · · ·(1 + |ξ|en
)−kn ,
• K (x) = η(x) +∑
S
∑J∈Γ(ES )∩Ns 2Q1 j1+···+Qs jsϕS
J (2j1 · xI1 , . . . , 2js · xIs )
as above, or even with the additional condition suppϕSJ ⊂ B.
![Page 58: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/58.jpg)
Equivalent characterizations of P0(E)
TheoremLet E be a matrix satisfying the basic assumptions. The following are
equivalent for K ∈ S ′(Rn):
• K ∈ P0(E),
• m = K is a smooth function satisfying the inequalities∣∣∂kξm(ξ)
∣∣ ≤ CkN(1 + |ξ|e1
)−k1 · · ·(1 + |ξ|en
)−kn ,
• K (x) = η(x) +∑
S
∑J∈Γ(ES )∩Ns 2Q1 j1+···+Qs jsϕS
J (2j1 · xI1 , . . . , 2js · xIs )
as above, or even with the additional condition suppϕSJ ⊂ B.
![Page 59: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/59.jpg)
Equivalent characterizations of P0(E)
TheoremLet E be a matrix satisfying the basic assumptions. The following are
equivalent for K ∈ S ′(Rn):
• K ∈ P0(E),
• m = K is a smooth function satisfying the inequalities∣∣∂kξm(ξ)
∣∣ ≤ CkN(1 + |ξ|e1
)−k1 · · ·(1 + |ξ|en
)−kn ,
• K (x) = η(x) +∑
S
∑J∈Γ(ES )∩Ns 2Q1 j1+···+Qs jsϕS
J (2j1 · xI1 , . . . , 2js · xIs )
as above, or even with the additional condition suppϕSJ ⊂ B.
![Page 60: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/60.jpg)
Properties (on Rn)
• Convolution by kernels in P0(E) is bounded on Lp, 1 < p <∞.
• P0(E) is closed under convolution.
• For E, E′ both satisfying the basic assumptions, the following areequivalent:
• P0(E) ⊂ P0(E′),• Γ(E) ⊂ Γ(E′),• for each j, k , e(j, k) ≤ e′(j, k).
• In particular, proper CZ kernels of type a are in P0(E) if and only if
a ∈ Γ(E).
• Given finitely many proper CZ kernels Ki of type ai , their convolution
belongs to P0(E), where
e(j, k) = maxi
aik
aij.
![Page 61: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/61.jpg)
Properties (on Rn)
• Convolution by kernels in P0(E) is bounded on Lp, 1 < p <∞.
• P0(E) is closed under convolution.
• For E, E′ both satisfying the basic assumptions, the following areequivalent:
• P0(E) ⊂ P0(E′),• Γ(E) ⊂ Γ(E′),• for each j, k , e(j, k) ≤ e′(j, k).
• In particular, proper CZ kernels of type a are in P0(E) if and only if
a ∈ Γ(E).
• Given finitely many proper CZ kernels Ki of type ai , their convolution
belongs to P0(E), where
e(j, k) = maxi
aik
aij.
![Page 62: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/62.jpg)
Properties (on Rn)
• Convolution by kernels in P0(E) is bounded on Lp, 1 < p <∞.
• P0(E) is closed under convolution.
• For E, E′ both satisfying the basic assumptions, the following areequivalent:
• P0(E) ⊂ P0(E′),• Γ(E) ⊂ Γ(E′),• for each j, k , e(j, k) ≤ e′(j, k).
• In particular, proper CZ kernels of type a are in P0(E) if and only if
a ∈ Γ(E).
• Given finitely many proper CZ kernels Ki of type ai , their convolution
belongs to P0(E), where
e(j, k) = maxi
aik
aij.
![Page 63: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/63.jpg)
Properties (on Rn)
• Convolution by kernels in P0(E) is bounded on Lp, 1 < p <∞.
• P0(E) is closed under convolution.
• For E, E′ both satisfying the basic assumptions, the following areequivalent:
• P0(E) ⊂ P0(E′),
• Γ(E) ⊂ Γ(E′),• for each j, k , e(j, k) ≤ e′(j, k).
• In particular, proper CZ kernels of type a are in P0(E) if and only if
a ∈ Γ(E).
• Given finitely many proper CZ kernels Ki of type ai , their convolution
belongs to P0(E), where
e(j, k) = maxi
aik
aij.
![Page 64: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/64.jpg)
Properties (on Rn)
• Convolution by kernels in P0(E) is bounded on Lp, 1 < p <∞.
• P0(E) is closed under convolution.
• For E, E′ both satisfying the basic assumptions, the following areequivalent:
• P0(E) ⊂ P0(E′),• Γ(E) ⊂ Γ(E′),
• for each j, k , e(j, k) ≤ e′(j, k).
• In particular, proper CZ kernels of type a are in P0(E) if and only if
a ∈ Γ(E).
• Given finitely many proper CZ kernels Ki of type ai , their convolution
belongs to P0(E), where
e(j, k) = maxi
aik
aij.
![Page 65: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/65.jpg)
Properties (on Rn)
• Convolution by kernels in P0(E) is bounded on Lp, 1 < p <∞.
• P0(E) is closed under convolution.
• For E, E′ both satisfying the basic assumptions, the following areequivalent:
• P0(E) ⊂ P0(E′),• Γ(E) ⊂ Γ(E′),• for each j, k , e(j, k) ≤ e′(j, k).
• In particular, proper CZ kernels of type a are in P0(E) if and only if
a ∈ Γ(E).
• Given finitely many proper CZ kernels Ki of type ai , their convolution
belongs to P0(E), where
e(j, k) = maxi
aik
aij.
![Page 66: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/66.jpg)
Properties (on Rn)
• Convolution by kernels in P0(E) is bounded on Lp, 1 < p <∞.
• P0(E) is closed under convolution.
• For E, E′ both satisfying the basic assumptions, the following areequivalent:
• P0(E) ⊂ P0(E′),• Γ(E) ⊂ Γ(E′),• for each j, k , e(j, k) ≤ e′(j, k).
• In particular, proper CZ kernels of type a are in P0(E) if and only if
a ∈ Γ(E).
• Given finitely many proper CZ kernels Ki of type ai , their convolution
belongs to P0(E), where
e(j, k) = maxi
aik
aij.
![Page 67: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/67.jpg)
Properties (on Rn)
• Convolution by kernels in P0(E) is bounded on Lp, 1 < p <∞.
• P0(E) is closed under convolution.
• For E, E′ both satisfying the basic assumptions, the following areequivalent:
• P0(E) ⊂ P0(E′),• Γ(E) ⊂ Γ(E′),• for each j, k , e(j, k) ≤ e′(j, k).
• In particular, proper CZ kernels of type a are in P0(E) if and only if
a ∈ Γ(E).
• Given finitely many proper CZ kernels Ki of type ai , their convolution
belongs to P0(E), where
e(j, k) = maxi
aik
aij.
![Page 68: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/68.jpg)
Homogeneous nilpotent groups
Let g be a homogeneous Lie algebra with (automorphic) dilations δ(t), t > 0.
Let
g = h1 ⊕ h2 ⊕ · · · ⊕ hm ,
be a decomposition into homogeneous subspaces, with the property that
[g, hj ] ⊂ hj+1 ⊕ · · · ⊕ hm for every j .
Call G the associated homogenous group, parametrized by g with the
Campbell-Hausdorff product. Denote by | | a homogeneous norm on G.
![Page 69: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/69.jpg)
Other homogeneities
Besides the standard CZ kernels adapted to the given dilations, we consider
CZ kernels adapted to other (non-automorphic) dilation
δa(t)x =(δ(t1/a1 )x1, . . . , δ(t1/am )xm
).
It is known (R.-Stein) that such kernels give bounded operators on Lp for
1 < p <∞, but in general they form a class which is not closed under
convolution.
TheoremSuppose that a1 ≤ a2 ≤ · · · ≤ am. Then the class of proper CZ kernels
adapted to the dilations δa is closed under convolution.
![Page 70: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/70.jpg)
Remarks
1. Let
K = η +∑j∈N
ϕ(j)j , K ′ = η′ +
∑j∈N
ψ(j)j
be the dyadic decompositions of two proper CZ kernels adapted to the
δa-dilations. Here ϕ(j)j denotes the bump function ϕj scaled by a factor 2−j ,
i.e. supported where
|x1| < 2−j/a1 , |x2| < 2−j/a2 , . . . |xm| < 2−j/am .
The monotonicity condition on the aj guarantees that the convolution of two
terms ϕ(j)j and ψ(j′)
j′ is scaled by a factor max{2−j , 2−j′}, like in RN .
2. For a specific homogeneous group, weaker assumptions may suffice. E.g.,
on H1 with the standard parabolic dilations and h1 = Rx , h2 = Ry , h3 = Rt ,
the condition 2/a3 ≤ 1/a1 + 1/a2 is enough.
![Page 71: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/71.jpg)
Convolution within P0(E) - Lp-boundedness
We say that E is doubly monotonic if
e(j, k) ≤ e(j − 1, k) , e(j, k) ≤ e(j, k + 1)
for all j, k .
TheoremAssume that E satisfies the basic assumptions and is doubly monotonic.
Then
• P0(E) is closed under convolution;
• convolution by kernels in P0(E) is bounded on Lp, 1 < p <∞.
![Page 72: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/72.jpg)
Convolution within P0(E) - Lp-boundedness
We say that E is doubly monotonic if
e(j, k) ≤ e(j − 1, k) , e(j, k) ≤ e(j, k + 1)
for all j, k .
TheoremAssume that E satisfies the basic assumptions and is doubly monotonic.
Then
• P0(E) is closed under convolution;
• convolution by kernels in P0(E) is bounded on Lp, 1 < p <∞.
![Page 73: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/73.jpg)
Convolution within P0(E) - Lp-boundedness
We say that E is doubly monotonic if
e(j, k) ≤ e(j − 1, k) , e(j, k) ≤ e(j, k + 1)
for all j, k .
TheoremAssume that E satisfies the basic assumptions and is doubly monotonic.
Then
• P0(E) is closed under convolution;
• convolution by kernels in P0(E) is bounded on Lp, 1 < p <∞.
![Page 74: Algebras of singular integral operators on nilpotent groupsweb.mat.bham.ac.uk/lmsmidlands2016/docs/slides_ricci.pdf · Dilations and CZ theory on nilpotent groups In the classical](https://reader033.vdocuments.us/reader033/viewer/2022042914/5f4e11f9b24368621a3dec37/html5/thumbnails/74.jpg)
Convolution of dyadic terms
Double monotonicity of E guarantees that the group convolution of two dyadic
terms at scales 2−J , 2−J′ with J, J ′ ∈ Γ(E) is at scale 2−J ∨ 2−J′ .
But the major problem in the proof is that the product cancellations are not
preserved under convolution.
The hypothesis of double monotonicity gives however a form of “weak
cancellation” of the convolution of two dyadic terms, which is sufficient to
prove that the whole resulting dyadic sum converges to a kernel in P0(E).