# MA4J0 Advanced Real Analysis

**Lecturer:** Filip Rindler

**Term(s):** Term 2

**Status for Mathematics students:** List C

**Commitment:** 30 lectures

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

**Formal registration prerequisites: **None

**Assumed knowledge: **

- MA3G7 Functional Analysis I: Banach spaces, Lebesgue spaces, dual spaces, linear operators.
- MA359 Measure Theory: General measures, Lebesgue measure & integral, properties of Lebesgue integral, convergence theorems.

**Useful background: **

- MA3G8 Functional Analysis II: More familiarity with Banach spaces and their main theorems.
- MA433 Fourier Analysis: Fourier transform & its properties.

**Synergies: **

- MA4A2 Advanced PDEs: Applications & motivation and graduate courses in Analysis.

**Content**: The module builds upon modules from the second and third year like MA222 Metric Spaces, MA359 Measure Theory and MA3G7 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 University Press, 1970.