#### RG Analysis and Partial Differential Equations

## V4B5: Real and harmonic analysis

#### Summer Semester 2018

- Dr. Pavel Zorin-Kranich
- Instructor
- Dr. Olli Saari
- Assistant

### Lectures

- Tu 14-16, 1.008
- Th 14-16, 1.008

### Exercise classes

- Fr 8-10, 1.008
- Fr 12-14, 0.011

### Topics

We have covered the following topics- (weak) L
^{p}spaces- Hölder, Minkowski, and Young convolution inequalities
- Banach spaces, duality, Hahn-Banach theorem (in extension form and in separation form)
- Dual spaces of L
^{p}spaces - Hardy-Littlewood maximal operator (weak type and L
^{p}estimates), Lebesgue differentiation - Real interpolation
- Complex interpolation (Riesz, Stein, iterated L
^{p}) - Hardy-Littlewood-Sobolev inequality, Sobolev embedding

- Fourier transform
- Distributions, Schwartz space, tempered distributions
- Action on Gaussians, multiplication formula, inversion formula, Plancherel theorem, Hausdorff-Young inequality
- Poisson summation formula
- Heisenberg uncertainty principle
- Hilbert transform

- Calderón-Zygmund theory
- CZ decomposition
- Cotlar-Stein lemma, cancellative CZ kernels
- Calderón-Vaillancourt theorem on pseudodifferential operators
- Norm convergence of Fourier integrals in dimension 1
- Cotlar's inequality
- Mihlin-Hörmander multipliers

- Littlewood-Paley theory
- Cauchy integral on Lipschitz curves
- Adapted Haar basis, almost orthogonal expansion in L
^{2} - Analytic capacity
- Bounded functions that are mapped to bounded functions by CZO

- Adapted Haar basis, almost orthogonal expansion in L
- Oscillatory integrals
- Fourier transform of surface-carried measures
- Lacunary spherical maximal function

- Hardy and BMO spaces
- Atomic decomposition
- Complex interpolation between H
^{1}and L^{p} - Spherical maximal theorem for d≥3
- H
^{1}-BMO duality - John-Nirenberg inequality
- Div-curl lemma
- Sharp maximal function
- Perturbation of constant coefficient elliptic PDE

- Ball multiplier
- Sequence valued extensions of linear operators
- Perron tree

### Lecture notes

These notes provide a record of which topics we discuss. Many details are omitted, but references are often provided.### Prerequisites

Lebesgue measure and integration, functional analysis (Banach spaces and operators).### Problem sets

- Exercises 1, due on 17 April.
- Exercises 2, due on 24 April.
- Exercises 3, due on 3 May.
- Exercises 4, due on 8 May.
- Exercises 5, due on 15 May.
- Exercises 6, due on 29 May.
- Exercises 7, due on 5 June.
- Exercises 8, due on 12 June.
- Exercises 9, due on 19 June.
- Exercises 10, due on 26 June.
- Exercises 11, due on 3 July.
- Exercises 12, due on 10 July.
- Exercises 13, due on 17 July.

### Literature

- E. M. Stein and R. Shakarchi, Functional analysis. Introduction to further topics in analysis. 2011.
- E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. 1993.
- Muscalu and Schlag, Classical and multilinear harmonic analysis, Vol 1. 2013
- Grafakos, Classical Fourier Analysis. 2008
- M. Christ, Lectures on singular integral operators. 1990
- Wolff, Lectures on harmonic analysis
- T. W. Körner, Fourier analysis (interesting historical notes)

## News

EMS Prize 2024 for Jessica Fintzen

Bonn mathematics performs excellently again in QS ranking

Stefan Schwede is invited speaker at the ECM 2024 in Sevilla

Jessica Fintzen wins Cole Prize

Regula Krapf receives university teaching award

Catharina Stroppel joined the North Rhine-Westphalia Academy for Sciences and Arts

Daniel Huybrechts receives the Compositio Prize for the periode 2017-2019

Catharina Stroppel receives Gottfried Wilhelm Leibniz Prize 2023

Grants for Mathematics students from Ukraine

Jessica Fintzen is awarded a Whitehead Prize of the London Mathematical Society

Peter Scholze elected as Foreign Member of the Royal Society