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Selected Topics in Harmonic Analysis (Winter Semester 2018/19)

The first lecture will be on the 22nd of October 2018.

In this Harmonic Analysis course we will deal mainly with Calderón-Zygmund and Singular integral operators. We will also study the BMO(\mathbb R^d) space, weighted norm inequalities-A_{\infty} weights and the nice interplay between BMO(\mathbb R^d) functions and the A_{\infty} class. Other topics we will cover are Reverse Hölder inequalities, the factorisation of A_{p} weights and the extrapolation theory of Rubio de Francia. If time permits we will also study how the Bellman function technique can be used to pass from dyadic operator estimates to Calderón-Zygmund operator estimates.

Lecture: Monday 14:00-15:30 SR 3.069


Hardy-Littlewood Maximal function, Calderón-Zygmund decomposition, Sharp maximal function and B.M.O., Singular Integral Operators, Multipliers, the A_{p} condition, Reverse Hölder Inequality and the A_{\infty} condition, Weighted norm inequalities for singular integrals, Factorisation of A_{p} weights and Extrapolation.


Familiarity with Measure theory, Lebesgue spaces, Fourier transform, Distributions and Functional Analysis


Oral exam (about 25 min).


David Cruz-Uribe, José Maria Martell, Carlos Pérez: Weights, Extrapolation and the Theory of Rubio de Francia, Birkhäuser Basel, 2011.

J. Garcia-Cuerva, J.L. Rubio de Francia: Weighted Norm Inequalities and Related Topics, North Holland, 1985.

Javier Duoandikoetxea: Fourier Analysis, Graduate Studies in Mathematics, AMS, 2001.

Elias M. Stein: Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.