Digital archive of theses discussed at the University of Pisa


Thesis etd-09012013-092700

Thesis type
Tesi di laurea magistrale
Thesis title
Multiparameter Singular Integrals: Product and Flag kernels.
Course of study
relatore Prof. Ricci, Fulvio
controrelatore Prof. Gueorguiev, Vladimir Simeonov
  • product kernel
  • Hilbert transform
  • CZO
  • Calderon-Zygmund
  • singular intergals
  • multiparameter
  • Flag kernel
Graduation session start date
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
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

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.