## Math 541 - Harmonic Analysis - Spring 2016

*Instructor*: Malabika Pramanik

*Office*: 214 Mathematics Building

*E-mail*: malabika at math dot ubc dot ca

*Lectures*: Mon,Wed,Fri 11 AM to 12 noon in Room 1118 Mathematics Annex Building.

*Office hours*: By appointment.

### Course information

### Piazza links

Piazza is a free, online question-and-answer platform for classes where you can ask questions to and receive answers from your classmates, anonymously if you wish, with some guidance from instructors. Use the signup link below to access Piazza. You will need a UBC email address for this.

### Week-by-week course outline

Here is a rough guideline of the course structure, arranged by week. The textbook sections are mentioned as a reference and as a reading guide. The treatment of these topics in lecture may vary somewhat from that of the text.

Depending on our actual progress in class, the weekly schedule may undergo minor adjustments. Stay tuned.

- Week 1: Some basic notions of real-variable theory
- Lebesgue density theorem
- Hardy-Littlewood maximal function
- Vitali covering lemma
- Distribution functions and Lebesgue norms

- Week 2: A brief foray into interpolation theorems
- Weak Lebesgue spaces
- Marcinkiewicz interpolation theorem

- Week 3: Interpolation, convolutions
- Riesz-Thorin interpolation theorem
- Three-lines lemma
- Convolution

- Week 4: Fourier transform and Plancherel's theorem
- Approximate identities
- Fourier transform and Fourier inversion
- Plancherel's theorem

- Week 5: Distributions
- Schwartz space of functions
- Fourier transform on the Schwartz space
- Tempered distributions
- Temperate measures

- Week 6: Distributions
- Principal value distribution
- Fourier transform on the space of distributions
- Applications: Kronecker's theorem

- Weeks 8 and 9: Applications of Fourier Analysis
- Evolution equations
- Isoperimetric inequality
- Harmonic conjugate and the Hilbert transform
- Norm convergence of Fourier series

- Weeks 10 and 11:
- Integral operator associated to a standard kernel
- Calderon-Zygmund theorem for singular integral operator
- Calderon-Zygmund decomposition of an integrable function

- Week 12
- Uniform convergence of Fourier series - yes and no.
- Kolmogorov's theorem on almost everywhere divergence.

### Homework