Introductory description

The module builds upon modules from the second and third year like Metric Spaces, Measure Theory and Functional Analysis I to present the fundamental tools in Harmonic Analysis and some applications, primarily in Partial Differential Equations.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Distributions on Euclidean space.
Tempered distributions and Fourier transforms.
Singular integral operators and Calderon-Zygmund theory.
Theory of Fourier multipliers.
Littlewood-Paley theory.

Learning outcomes

By the end of the module, students should be able to:

• Definition of Sobolev spaces using the Fourier Transform and the connections between the decay of the Fourier Transform and the regularity of functions.
• Connection between the boundedness of singular integrals and the existence of solutions for some partial differential equation.
• Connections between singular integrals and the summation of Fourier series.
• Understanding the scaling properties of inequalities.
• Reinforce analytical thinking developed in previous courses.

Indicative reading list

• Friedlander, G. and Joshi, M. : Introduction to the theory of distributions, 2nd edition, Cambridge University Press, 1998.
• Duoandikoetxea, J. : Fourier Analysis - American Mathematical Society, Graduate Studies in Mathematics, 2001.
• Muscalu C. and Schlag, W. : Classical and Multilinear Harmonic Analysis, Cambridge Studies in advanced Mathematics, 2013.
• Folland, G. Real Analysis: Modern Techniques and their applications, Wiley 1999.
• Grafakos, L. : Classical Fourier Analysis - Springer 2008.
• Grafakos, L.: Modern Fourier Analysis - Springer 2008.
• Stein, E.M.: Singular Integrals and differentiability properties of functions and differentiability properties of functions - Princeton Univesity Press, 1970.

Subject specific skills

1. Understand the calculus of distributions. In particular, see how one can differentiate non-smooth objects and solve differential equations without smooth (or even differentiable functions).
2. How to properly understand point 1 above within the context of duality.
3. Study boundedness of operators with singular kernels. In particular, one would like to understand Calderon-Zygmund decomposition in n dimensions (the one-dimensional case having been studied in term in Fourier analysis).
4. Analyse frequency decompositions into dyadic blocks. This is done within the context of Littlewood-Paley theory.
5. Analyse Fourier multiplier operators.
6. Analyse Sobolev spaces, in particular with degree of differentiation being a real number (the case of integer-derivative Sobolev spaces is covered in other modules).

Transferable skills

1. Understand how to apply rules of calculus in the most general context and get a new perspective on this.
2. Obtain `big picture' thinking from 1
3. Improve the writing of rigorous proofs.
4. Prepare students for a PhD in mathematical analysis or a related field.