# MA4J0 Content

**Content**: The module build upon modules on 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:

- Exploring the connections between singular integrals and the summation of Fourier series.
- Studying Connection between the boundedness of singular integrals and the existence of solutions for some partial differential equations.
- 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**:

- Fourier Series and advanced convergence topics.
- Fourier transform in : including - Schwartz space (including distribution theory). - Sobolev spaces (including Rellich's theorem and Sobolev embedding). - Littlewood-Paley (applications to PDE).
- Application to PDE. - Elliptic/Heat equation and the Poisson Kernel & motivation for the study singular integrals. - Wave equation.
- Maximal functions (Poisson kernel).
- and Marcinkiewicz interpolation.
- Other singular integrals (Hilbert transform - motivated by the multiplier theorem). Higher dimensional multiplier problems.
- Riesz potentials and ractional derivatives.

**Books**:

- 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.

- Duoandikoetxea, J. *Fourier Analysis* - American Mathematical Society 2001.