# 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 $\R^n$ : including - Schwartz space (including distribution theory). - $H^s$ 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).
• $L^p$ 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 functionsand differentiability properties of functions - Princeton Univesity Press.
- Duoandikoetxea, J. Fourier Analysis - American Mathematical Society 2001.