Calderon-Zygmund Theory of Singular Integrals:

Calderon-Zygmund Theory of Singular Integrals:

In their pioneering papers which appeared in American Journal of
Mathematics(1956) A. P. Calderon and A. Zygmund defined a class of kernels
that generalized the Hilbert transform and Riesz transform at the same time
to higher dimensional settings and they studied the L^p boundedness of
these kernels. Over the decades, Calderon-Zygmund theory has proven to be
an extremely useful toolbox for mathematicians not only working in pure
areas like operator theory but also in applied fields such as partial
differential equations. In this course this theory will be constructed and
examined in detail. Although parts of the course content may be generalized
to more abstract settings such as topological groups and manifolds, we will
confine ourselves to the Euclidean space R^n. A graduate student with a
solid background in Real Analysis(Integration and Measure Theory) should be
able to follow the course with no difficulty.

The Course Content:

0. Preliminaries:

(i) L^p spaces: Distribution function and its applications, weak type
operators on L^p spaces.
(ii)Interpolation of Operators on L^p spaces: Marcinkiewicz, Riesz-Thorin
and Stein Interpolation theorems
(iii)Some basic inequalities: Hölder, Minkowski and Hausdorff-Young
inequalities
(iv)Locally compact groups and Haar measure: Hausdorff-Young inequality on
locally compact groups, as an application Hardy's inequality
(v) Schwartz space & Fourier transform: L^2 theory, Plancherel Theorem,
Inversion formula, Fourier transform on locally compact groups
(vi) Maximal Functions: Hardy-Littlewood maximal operator, Lebesgue
differentiation theorem

1. Singular Integrals of Convolution Type:

(i) Hilbert transform: Definition and basic properties, L^p boundedness of
Hilbert transform, Connection with analytic functions, Poisson kernel,
conjugate Poisson kernel, Operator of harmonic conjugation, maximal Hilbert
transform
(ii) Riesz transform: Definition and basic properties, Riesz transform as a
Fourier multiplier, Connections with the Laplacian
(ii)Homogeneous Singular Integrals(Calderon-Zygmund kernels): Definition
and properties of Calderon-Zygmund kernels, L^p boundedness of Calderon-
Zygmund kernels, Calderon-Zygmund decomposition lemma

2. Singular Integrals of Non-convolution Type:

(i) Singular Integrals with variable kernels: Definition and basic
properties, L^p boundedness of Singular Integrals with variable kernels
(ii) Banach algebras of S?ngular Integral Operators