Topics in Harmonic Analysis

News

• Course website created.

Lecture times

• Lectures: Mondays and Wednesdays 10:15–12 in MaD 245
• Exercises: Tuesdays 8:30–10 in MaD 355
• First lecture: 23rd October
• First exercises: 31st October

Course description

The course will focus on singular integral operators (SIOs), which are operators of the form $Tf(x) = \int K(x,y) f(y) dy$, where the kernel $K(x,y)$ has a singularity on the diagonal $x=y$. The basic question we will study is: under what hypotheses on the kernel $K$ is the operator $T$ bounded on $L^p(\mathbb{R}^n)$? Questions of this type appear naturally e.g. in partial differential equations. We will cover a mix of classical results and more recent advances. Specific topics may include:

• classical examples (Hilbert transform, Riesz transform, Cauchy transform) and their connection to PDEs
• dyadic cubes and why they are useful
• general SIOs and the Calderón-Zygmund decomposition
• the Littlewood-Paley theory
• SIOs in the weighted setting, the Muckenhoupt $A_p$ weights
• sparse domination and the $A_2$ theorem of Hytönen
• the Cauchy transform on Lipschitz graphs

Materials

Lecture notes will be posted on this website. To get a flavour of the course’s contents, see

• chapters 4–9 in the lecture notes of Parissis,
• lecture notes of Pérez
• parts 3 and 4 of the lecture notes of Tao
• Stein’s Singular Integrals book
• the books of Grafakos: Classical Fourier Analysis and Modern Fourier Analysis

Exercise lists

The exercise lists will be posted here weekly.

Prerequisites

A good knowledge of measure and integration theory is necessary to follows the course. I also expect the students to know the definition and basic properties of the Fourier transform. Some basic functional analysis will be helpful, too. For example, it would be good if you knew the answer to these questions:

• If $\mu$ is a measure on $\mathbb{R}^n$, and $1 < p < \infty$, then what is the dual of $L^p(\mu)$?
• How can you use Plancherel’s theorem to define the Fourier transform on $L^2(\mathbb{R}^n$)?
• If $X$ and $Y$ are Banach spaces, and $T:X\to Y$ is a linear operator, what does it mean for $T$ to be bounded?

It is recommended to take the course in parallel with the Real Analysis course, unless one is already familiar with its contents. We will use some results covered in that class, e.g. the properties of the Hardy-Littlewood maximal function, or the Marcinkiewicz interpolation theorem.

Passing the course

There will be no grade other than “passed” or “failed.” To pass the course, you need to:

• solve at least 50% of the problems discussed at the weekly exercise sessions,
• at the end of the course, make a presentation on a chosen topic related to singular integral operators. Possible topics of presentations will be posted here at some point.