An Introduction to the Theory of Singular Integrals and the Ap Theory of Weights

Dates: 06/07/2015 - 10/07/2015 (5 sessions)
Time: 9:30 - 11:30 (a total of 10 hours)

In very recent years there has been a revival of the classical theory of Singular Integrals with weights. New questions have lead to different methods improving very well known classical results in a very surprising way. In this mini-course we plan to give a short introduction to some aspects of these new point of view. The goal is to study Singular Integral Operators on weighted Lp spaces with sharp bounds in terms of the Ap or related constant. We will start by considering first the case of the Hardy-Littlewood maximal function where we will review some aspects of the classical theory from a modern point of view: new sharp Reverse Holder inequalities, improvement of the Muckenhoupt-Buckley's theorem and the two weight case, factorization and extrapolation. Then we will explain and discuss how to pass from the continuous Singular Integrals to their corresponding discrete versions of and how to use it. Finally, if there is some time, the special A1 class will be considered.

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