# MA4J0 Advanced Real Analysis

**Lecturer:** Vedran Sohinger

**Term(s):** Term 2

**Status for Mathematics students:** List C

**Commitment:** 30 lectures

**Assessment:** 3 hour exam (100%).

**Prerequisites:** MA3G7 Functional Analysis I & MA359 Measure Theory. Desirable: MA3G8 Functional Analysis II, MA433 Fourier Analysis

**Content**: 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. Some of the main aims include:

- Setting up a rigorous calculus of rough objects, such as distributions.
- Studying the boundedness of singular integrals and their applications.
- Understanding the scaling properties of inequalities.
- Defining Sobolev spaces using the Fourier Transform and the connections between the decay of the Fourier Transform and the regularity of functions.

**Outline**:

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

**Books**:

- 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.