### Fall 2023

**Sagun Chanillo**

### Course Description:

Topics:

1. Interpolation theorems in generality of Marcinkiewicz, Riesz-Thorin and Stein.

2. Riesz potentials, Hardy-Littlewood-Sobolev fractional integration theorem. Poincare

and local Sobolev inequalities. Gagliardo-Nirenberg theorem, Moser-Trudinger inequality.

3. Calderon-Zygmund singular integrals the L

2

theory. The non-convolution case Cotlar’s

Lemma.

4. The L

p

theory of singular integrals, the Calderon-Zygmund decomposition.

5. Maximal singular integrals.

6. BMO, the Spanne-Stein theorem, the John-Nirenberg inequality.

7. The Fefferman-Stein sharp function.

8. H¨ormander’s multiplier theorem.

9. Khinchine’s inequality/ large deviation estimate and the Littlewood-Paley theorem.

10. Restriction theory of the Fourier transform.

11. Bochner-Riesz theorem of Fefferman and Strichartz inequality.

12. The theory of Ap weights and Muckenhoupt’s theory for the Hardy-Littlewood maximal

function and weighted inequalities for singular integrals.

13. (Time permitting) Carleson measures and Fefferman’s theorem that BMO is the dual

of Hardy space.

### Text:

There is no textbook for this course. Lecture notes for each Lecture will be

posted on Canvas.

### Prerequisites:

TBA

### ***********************************************************

### Fall 2022

**Mariusz Mirek**

### Course Description:

In the first part of the course, we will study basic concepts in Harmonic Analysis, specifically, interpolation theorems, Hardy-Littlewood maximal function and the Hardy-Littlewood-Sobolev fractional integration theorem. This will be followed by the Littlewood-Paley theory and their applications in r-variational estimates as well as the Calderon-Zygmund theory of singular integrals and Radon operators. In the second part of the course we will study restriction theorems and Kakeya maximal functions, we will also construct the Besicovitch set. This will be a starting point of the decoupling theory, which will be used to prove the Vinogadov mean value conjecture from number theory. Finally, we will prove the Carleson theorem, which asserts that partial sums of Fourier series of a square integrable function converge pointwise almost everywhere.

### Text:

No textbook

### Prerequisites:

501, 502

### ***********************************************************

### Fall 2020

**Sagun Chanillo**

### Course Description:

This course is a basic course on Harmonic Analysis. We will study, interpolation theorems the Hardy-Littlewood-Sobolev fractional integration theorem. Then Calderon-Zygmund theory of Singular integrals. After that we will prove the Hormander multiplier theorem. This will be followed by Littlewood-Paley theory. We will end with the study of Fourier transform restriction theorems, applications to the Strichartz estimates for wave and Schrodinger equations and the theory of Bochner-Riesz multipliers. This course is addressed to students who need these tools in Nonlinear analysis and PDE and in their study of elliptic, parabolic and hyperbolic problems.

### Text:

None

### Prerequisites:

Math 501, 502, 503.

### Schedule of Sections: