## Harmonic Analysis

אנליזה הרמונית

### Spring semester, 2020

: Bo'az Klartag**Lecturer**

Rotem Assouline**Teaching assistant:**

: Monday, 14:15 - 17:00.**Classes**The first class is scheduled for April 20, and will be given online using zoom. In order to join the meeting, use this link (or alternatively, type in zoom the meeting id: 443-524-154).

The semester at Weizmann begins on April 19, 2020 and ends on July 24, 2020. Updates should appear in the graduate school calendar.

: Please fill this form in order to join the course mailing list. Students that plan to submit their homework must join the mailing list.**Mailing list**

__Syllabus__Fourier transform in Euclidean space, tempered distributions, singular integrals, stationary phase, pseudodifferential operators, elliptic regularity, uncertainty principle, Weyl law.

**Prerequisites**Familiarity with multivariate calculus (say, the divergence threorem) and real analysis (say, the Lebesgue measure).

**Final grade**It will be based on the solution of the homework exercises.

**Related literature**

- H. Dym, H. P. McKean, Fourier Series and Integrals.
- Y. Katznelson, An introduction to Harmonic Analysis.
- T. Körner, Fourier Analysis.
- E. Stein, Singular Integrals and Differentiability Properties of Functions.
- E. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces.
- X. Saint Raymond, Elementary introduction to the theory of pseudodifferential operators.

**Lecture notes**

- Lecture 1: Measures in R^n, Hausdorff dimension, Frostman Lemma.
- Lecture 2: Fourier transform in R^n, Schwartz functions, Inversion formula.
- Lecture 3: Inversion formula, Plancherel, Uncertainty principle, a bit about Radon transform.
- Lecture 4: Radon transform, Kakeya sets in 2D are of full dimension.
- Lecture 5: Duality in Functional Analysis, tempered distributions.
- Lecture 6: Construction of Kakeya set, Fourier transform of tempered distributions and pv(1/x).
- Lecture 7: Homogeneous tempered distributions, applications to projections of fractals.
- Lecture 8: Littlewood-Paley decomposition, Bernstein's lemma.
- Lecture 9: Holder norm through Fourier transform, elliptic regularity for the Laplacian, Morrey's lemma.
- Lecture 10: Hilbert transform, introduction to pseudo-differential operators, Cotlar-Stein lemma.
- Lecture 11: Calderon-Vaillancourt theorem, i.e., boundness in L^2 and Sobolev spaces of psido of order 0. Continuity in Schwartz topology of any psido.
- Lecture 12: Symbol calculus of psido's.
- Lecture 13: Pseudo-locality and preservation of regularity, parametrix for elliptic psido, Holder class is preserved by psido of order 0.

Video recordings of the lectures should be available here.

**Home assignments**

- Homework assignment no. 1, and a proposed solution by Matan Eilat.
- Homework assignment no. 2.
- Homework assignment no. 3.
- Homework assignment no. 4.