# Topics in Harmonic Analysis

Singular Integral Operators

## News

- Final version of lecture notes uploaded.
- Lecture notes updated.
- Fifth exercise list added, lecture notes updated.
- Fourth exercise list added, lecture notes updated.
- Presentation topics added.
- Third exercise list added, lecture notes updated.
- Second exercise list added, lecture notes updated.
- First exercise list and the lecture notes for the first lectures added.
- 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
- Last lecture: 7th December at 8:30 in MaD 355
- First exercises: 31st October
- Last exercises: 28th November
- Presentations: 12th December at 8:30 in MaD 355

## Materials

Lecture notes are updated weekly. You can send me an e-mail if you spot typos. If you want to read more on the topic, see

- the book of Stein: Singular Integrals
- the books of Grafakos: Classical Fourier Analysis and Modern Fourier Analysis
- the book of Duoandikoetxea: Fourier Analysis
- the lecture notes of Parissis
- the lecture notes of Tao

## Exercise lists

- 1st exercise list, solutions by Carmelo
- 2nd exercise list, solutions by Carmelo
- 3rd exercise list, solutions by Carmelo
- 4th exercise list, solutions by Carmelo
- 5th exercise list, solutions by Carmelo

The teaching assistant is Carmelo Puliatti. If you can’t make it to class you can e-mail him the solutions, his address is carmelo.c.puliatti [at] jyu.fi.

## 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, prepare a presentation and a short “essay” (3-5 pages) on a chosen topic related to singular integral operators. Possible topics of presentations are listed below.

## Presentation topics

If you want to book a topic, send me an e-mail or tell me during a lecture. Feel free to ask more questions about the proposed topic; it is also possible to propose another topic related to SIOs which is not listed below.

~~connection of Hilbert transform with analytic functions, Poisson and conjugate Poisson kernels (Duoandikoetxea sections 3.1 and 3.2, or “Classical” Grafakos section 5.1.2)~~-**booked by Chengxi, presentation on 12.12 at 8:30 in MaD 355**~~application of Hilbert transform to the convergence of partial Fourier integrals (lecture notes of Parissis section 5.3.2)~~-**booked by Jani, presentation on 12.12 at 8:30 in MaD 355**~~sufficient conditions on a kernel that ensure $L^2$ boundedness of a SIO (Duoandikoetxea sections 4.1 and 4.2, or Duoandikoetxea section 5.2, or “Classical” Grafakos section 5.4.1)~~-**booked by Guangzeng, presentation on 12.12 at 8:30 in MaD 355**- method of rotations (Duoandikoetxea section 4.3, or “Classical” Grafakos section 5.2.3)
- endpoint estimates 1: the Hardy space and SIOs applied to H^1 functions (Duoandikoetxea section 6.1, or “Modern” Grafakos Theorem 4.2.6)
- endpoint estimates 2: the space BMO and SIOs applied to bounded functions (Duoandikoetxea section 6.2, or “Modern” Grafakos section 4.1.3 and Theorem 4.2.7)
- commutators of SIOs (lecture notes of Parissis, section 7.4). Requires coordination with the BMO presentation.

## 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:

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

## 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.