## Thesis etd-09012013-092700 |

Thesis type

Tesi di laurea magistrale

Author

URALTSEV, GENNADY

URN

etd-09012013-092700

Thesis title

Multiparameter Singular Integrals: Product and Flag kernels.

Department

MATEMATICA

Course of study

MATEMATICA

Supervisors

**relatore**Prof. Ricci, Fulvio

**controrelatore**Prof. Gueorguiev, Vladimir Simeonov

Keywords

- product kernel
- Hilbert transform
- CZO
- Calderon-Zygmund
- singular intergals
- multiparameter
- Flag kernel

Graduation session start date

16/09/2013

Availability

Full

Summary

This thesis is concerned with the study of multi-parameter

singular integrals on the Euclidean space. The Schwartz Kernel Theorem

states that translation invariant continuous linear operators with

minimal smoothness conditions are convolution operators. Singular

integral operator theory is concerned with the study of the singular

kernels associated with such operators. A well developed theory exists

for the class of Calderón-Zygmund operators and the associated kernels.

This kind of kernels can be seen as a natural generalization of the

Hilbert kernel on $\R$, of the Riesz kernels in $\R^n$, and, more

generally, of kernels of homogeneous degree $-n$ in $\R^n$.

Calderón-Zygmund theory is a one-parameter homogeneous theory

since the kernels of interest are well-behaved with respect to a family

of homogeneous dilation with one parameter. Calderón-Zygmund kernels

arise from many problems in linear PDEs and complex analysis. $L^p$

boundedness for $p\in (1,+\infty)$ and stability under composition are

well known results for such kernels.

Product-type kernels arise naturally in analysis in several complex

variables and PDEs. As a matter of fact joint spectral functional

calculus for more than one differential operator naturally produce to

product structures. Product spaces occur naturally in the heat equation

or in the Shrödinger equation.

Abstractly, product kernels are the result of the extension of

Calderón-Zygmund theory to product spaces. The tensor product of two or

more Calderón-Zygmund kernels gives a singular kernel defined on the

product space. The new kernel has a singularity not only in the origin

but also along all coordinate sub-spaces. From the point of view of the

associated operators, the tensor product corresponds to the composition

of the original operators acting independently on the coordinates of the

product space. Product kernel theory aims to extend the space of tensor

products of Calderón-Zygmund kernels to a suitably defined completion.

This is done mainly by using multi-parameter dilation techniques, with

one parameter for each factor of the product space. An other idea that is

pursued is that product theory can be inspired by vector valued

functional analysis and integration. While avoiding a too abstract

approach to such functional analysis in this thesis, some ideas

are shown to be very useful.

This thesis illustrates the adaptation of some important results inspired

by Calderón-Zyg-mund theory to product kernels. These include

decomposing

a kernel into a multi-parameter dyadic series of homogeneous dilates of

smooth functions concentrated on essentially disjoint scales and,

conversely, finding conditions when such dyadic sums converge to product

kernels. Furthermore, since tensor products of bounded operators on $L^p$

remain bounded on $L^p$ one can suppose that this remains true for

general product kernels. However, the proof usually used for

Calderón-Zygmund operators does not seem to be generalized to

product kernels since weak $L^1-L^{1}_{w} $ boundedness fails. A finer

technique based on product square function estimates and product

Littlewood-Payley theory is developed to solve this problem.

This idea is based on the quasi-orthogonality of the dyadic decomposition

for the kernels.

The second part of this thesis deals with a certain sub-class of product

kernels given by flag kernels. While product kernels are the most

intuitive generalization of Calderón-Zygmund theory to a multi-parameter

setting, the singularities are generally too many to work with directly.

Flag kernels have singularities concentrated on a flag or filtration of

the space, and not along all coordinate subspaces. Kernels with such an

ordered structure of singularities appear more often from concrete

problems than general product kernels. A multi-parameter theory for flag

kernels similar to the one for product kernels is developed. We also show

that even though flag kernels form a sub-class of product kernels any

product kernel can be written as a sum of flag kernels adapted to

different flags.

These results were already present in literature. Flag

kernels were introduced by Nagel,

Ricci, and Stein in “Singular integrals with flag kernels and analysis on quadratic CR mani-

folds”, Journal of Functional Analysis, 2001.

A large portion of the above paper is dedicated to applications of product-type singular integral operators. Here we develop the results and

provide detailed proofs based on the ideas contained in the part of that paper dedicated to the general theory of flag kernels.

In this thesis we also establish several new results. While the question

of whether changes of variables conserve product and flag kernels will be

addressed in a forthcoming paper by Alexander Nagel, Fulvio Ricci, Elias

Stein, and Richard Wainger, they deal only with polynomial changes of

variable. We show that the classes of Calderón-Zygmund, product and flag

kernels with compact support are stable with respect to generic smooth

changes of variable that have the geometric property of fixing the

singular subspaces. These results and the techniques we use can be the

first step to studying product-type singular integral operators on

manifolds.

Furthermore an attempt is made to develop a basic functional calculus for

product singular integrals with respect to derivation,

multiplication and convolution. This is done by introducing kernels of

generic pseudo-differential order. We establish some useful facts but

show that some properties may fail except for a restricted range of

pseudo-differential orders.

Finally we show how this functional calculus can be used to establish an

approximation result for kernels composed with changes of variable.

singular integrals on the Euclidean space. The Schwartz Kernel Theorem

states that translation invariant continuous linear operators with

minimal smoothness conditions are convolution operators. Singular

integral operator theory is concerned with the study of the singular

kernels associated with such operators. A well developed theory exists

for the class of Calderón-Zygmund operators and the associated kernels.

This kind of kernels can be seen as a natural generalization of the

Hilbert kernel on $\R$, of the Riesz kernels in $\R^n$, and, more

generally, of kernels of homogeneous degree $-n$ in $\R^n$.

Calderón-Zygmund theory is a one-parameter homogeneous theory

since the kernels of interest are well-behaved with respect to a family

of homogeneous dilation with one parameter. Calderón-Zygmund kernels

arise from many problems in linear PDEs and complex analysis. $L^p$

boundedness for $p\in (1,+\infty)$ and stability under composition are

well known results for such kernels.

Product-type kernels arise naturally in analysis in several complex

variables and PDEs. As a matter of fact joint spectral functional

calculus for more than one differential operator naturally produce to

product structures. Product spaces occur naturally in the heat equation

or in the Shrödinger equation.

Abstractly, product kernels are the result of the extension of

Calderón-Zygmund theory to product spaces. The tensor product of two or

more Calderón-Zygmund kernels gives a singular kernel defined on the

product space. The new kernel has a singularity not only in the origin

but also along all coordinate sub-spaces. From the point of view of the

associated operators, the tensor product corresponds to the composition

of the original operators acting independently on the coordinates of the

product space. Product kernel theory aims to extend the space of tensor

products of Calderón-Zygmund kernels to a suitably defined completion.

This is done mainly by using multi-parameter dilation techniques, with

one parameter for each factor of the product space. An other idea that is

pursued is that product theory can be inspired by vector valued

functional analysis and integration. While avoiding a too abstract

approach to such functional analysis in this thesis, some ideas

are shown to be very useful.

This thesis illustrates the adaptation of some important results inspired

by Calderón-Zyg-mund theory to product kernels. These include

decomposing

a kernel into a multi-parameter dyadic series of homogeneous dilates of

smooth functions concentrated on essentially disjoint scales and,

conversely, finding conditions when such dyadic sums converge to product

kernels. Furthermore, since tensor products of bounded operators on $L^p$

remain bounded on $L^p$ one can suppose that this remains true for

general product kernels. However, the proof usually used for

Calderón-Zygmund operators does not seem to be generalized to

product kernels since weak $L^1-L^{1}_{w} $ boundedness fails. A finer

technique based on product square function estimates and product

Littlewood-Payley theory is developed to solve this problem.

This idea is based on the quasi-orthogonality of the dyadic decomposition

for the kernels.

The second part of this thesis deals with a certain sub-class of product

kernels given by flag kernels. While product kernels are the most

intuitive generalization of Calderón-Zygmund theory to a multi-parameter

setting, the singularities are generally too many to work with directly.

Flag kernels have singularities concentrated on a flag or filtration of

the space, and not along all coordinate subspaces. Kernels with such an

ordered structure of singularities appear more often from concrete

problems than general product kernels. A multi-parameter theory for flag

kernels similar to the one for product kernels is developed. We also show

that even though flag kernels form a sub-class of product kernels any

product kernel can be written as a sum of flag kernels adapted to

different flags.

These results were already present in literature. Flag

kernels were introduced by Nagel,

Ricci, and Stein in “Singular integrals with flag kernels and analysis on quadratic CR mani-

folds”, Journal of Functional Analysis, 2001.

A large portion of the above paper is dedicated to applications of product-type singular integral operators. Here we develop the results and

provide detailed proofs based on the ideas contained in the part of that paper dedicated to the general theory of flag kernels.

In this thesis we also establish several new results. While the question

of whether changes of variables conserve product and flag kernels will be

addressed in a forthcoming paper by Alexander Nagel, Fulvio Ricci, Elias

Stein, and Richard Wainger, they deal only with polynomial changes of

variable. We show that the classes of Calderón-Zygmund, product and flag

kernels with compact support are stable with respect to generic smooth

changes of variable that have the geometric property of fixing the

singular subspaces. These results and the techniques we use can be the

first step to studying product-type singular integral operators on

manifolds.

Furthermore an attempt is made to develop a basic functional calculus for

product singular integrals with respect to derivation,

multiplication and convolution. This is done by introducing kernels of

generic pseudo-differential order. We establish some useful facts but

show that some properties may fail except for a restricted range of

pseudo-differential orders.

Finally we show how this functional calculus can be used to establish an

approximation result for kernels composed with changes of variable.

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