Mathematical Institute of the University of Bonn

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

1. (weak) L^p spaces
   1. Hölder, Minkowski, and Young convolution inequalities
   2. Banach spaces, duality, Hahn-Banach theorem (in extension form and in separation form)
   3. Dual spaces of L^p spaces
   4. Hardy-Littlewood maximal operator (weak type and L^p estimates), Lebesgue differentiation
   5. Real interpolation
   6. Complex interpolation (Riesz, Stein, iterated L^p)
   7. Hardy-Littlewood-Sobolev inequality, Sobolev embedding
2. Fourier transform
   1. Distributions, Schwartz space, tempered distributions
   2. Action on Gaussians, multiplication formula, inversion formula, Plancherel theorem, Hausdorff-Young inequality
   3. Poisson summation formula
   4. Heisenberg uncertainty principle
   5. Hilbert transform
3. Calderón-Zygmund theory
   1. CZ decomposition
   2. Cotlar-Stein lemma, cancellative CZ kernels
   3. Calderón-Vaillancourt theorem on pseudodifferential operators
   4. Norm convergence of Fourier integrals in dimension 1
   5. Cotlar's inequality
   6. Mihlin-Hörmander multipliers
4. Littlewood-Paley theory
5. Cauchy integral on Lipschitz curves
   1. Adapted Haar basis, almost orthogonal expansion in L²
   2. Analytic capacity
   3. Bounded functions that are mapped to bounded functions by CZO
6. Oscillatory integrals
   1. Fourier transform of surface-carried measures
   2. Lacunary spherical maximal function
7. Hardy and BMO spaces
   1. Atomic decomposition
   2. Complex interpolation between H¹ and L^p
   3. Spherical maximal theorem for d≥3
   4. H¹-BMO duality
   5. John-Nirenberg inequality
   6. Div-curl lemma
   7. Sharp maximal function
   8. Perturbation of constant coefficient elliptic PDE
8. Ball multiplier
   1. Sequence valued extensions of linear operators
   2. 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)