Harmonic Analysis

Prerequisites:

Measure Theory and Real Analysis

Course Description:

The primary purpose of this course is to give students a presentation of the basic materials in singular integrals and related topics in harmonic analysis. The course is designed for graduate students as well as advanced undergraduate students in mathematics.

Course Content:

This is an introductory course to singular integrals and related topics in harmonic analysis. The topics that will be covered in this course include

1. L^p and weak L^p spaces, convolution, interpolation theorems, Hardy-Littlewood maximal functions, and Fourier transform.

2. Hilbert transform and Riesz transforms, Calderon-Zygmund decomposition, Calderon-Zygmund theorems, and applications to partial differential equations (PDEs).

3. A real-variable method and its applications to PDEs.

4. Hardy and BMO spaces.

5. The theory of A_p weights.

6. Fourier multiplier and Littlewood-Paley theory, almost orthogonality, L^2 theory of operators with Calderon-Zygmund kernels, the Cauchy integral and layer potentials, and applications to boundary value problems.

References:

Fourier Analysis by Javier Duoandikoetxea

Harmonic Analysis by Elias Stein

Classical and Modern Fourier Analysis by Loukas Grafakos