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.
- 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.
- 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.