johanna.mcbryde@icms.org.uk      0131 650 9816

THE OFFICIAL WEBPAGE OF THIS COURSE IS http://www.macs.hw.ac.uk/~hg94/ha17 .

All relevant information, lecture notes etc. will be posted there.

 

 

Aim: This course aims to give an introduction to harmonic analysis and discuss related standard techniques in real analysis, linear PDE and function spaces.

Prerequisites: SMSTC stream "Pure Analysis 1" or equivalent

Contents:
Chapter 1: Preliminaries - Distributions and Fourier Transform [content: test functions, Schwartz space,
distributions, Fourier transform, Marcinkiewicz and Riesz-Thorin interpolation theorems, Paley-Wiener
theorem for compactly supported distributions]
Chapter 2: Introduction to Integral Operators [content: examples, Schur test, truncation theorems: smooth truncations and Christ-Kiselev, maximal operators]
Chapter 3: Hardy-Littlewood Maximal Inequality and Covering Theorems [content: Vitali-type covering lemma, Calderon-Zygmund decomposition, Hardy-Littlewood maximal inequality, Lebesgue differentiation theorem, complement: TT* proof]
Chapter 4: Calderon-Zymund Theory [content: singular integral kernels, Calderon-Zygmund theorem]
Chapter 5: Pseudodifferential Operators, Elliptic Theory, Function Spaces [content: pseudodifferential operators on Rn, composition, boundedness, microlocal parametrix construction, Sobolev spaces, their Littlewood-Paley characterisation, Besov-Triebel-Lizorkin spaces]
Chapter 6 (we might not get there): Fourier Analysis on Locally Compact Abelian Groups [content: basic definitions and theorems up to inversion theorem for the Fourier transform and the Plancherel Theorem]

Main references:
* own lecture notes
* T. Tao, Graduate Fourier Analysis, lecture notes UCLA
* G. Grubb, Distributions and Operators, Springer Graduate Texts in Mathematics
* L. Hörmander, The Analysis of Linear Partial Differential Operators I, Springer Grundlehren
* W. Rudin, Fourier Analysis on Groups, Interscience Publishers

Remark: See http://www.macs.hw.ac.uk/~hg94/dif12 for a similar previous course.

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Tel : 0131 650 9816

Fax : 0131 651 4381
Mail : johanna.mcbryde@icms.org.uk