Minicourses

A short introduction to the (modern) theory of Dirichlet series

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The main goal of the lectures is to describe differentiability properties of measurable functions defined in the euclidean space using square functions which involve their  second symmetric divided differences.  Classical results of Marcinkiewicz, Stein and Zygmund will be discussed as well as their discrete versions which can be stated in terms of dyadic martingales. Results of Burkholder, Gundy and Stout relating the assymptotic behavior of a dyadic martingale with the size of its quadratic variation will also be discussed. An averaging argument will be used to transfer the Fatou type results and the Law of the Iterated Logarithm which hold in the discrete setting to the continuous one.

Commutators (or to an operator theorist, Hankel operators) and the standard class of operators considered in Harmonic Analysis, Calder\’on-Zygmund operators, exhibit different behaviours, including differences of the optimal bounds in a weighted setting, which are surprising at a first glance. We will start the course by introducing the setting of vector-valued functions and matrix-weighted weight and presenting some of the recent progress in the area. We will then show that the matrix-weighted setting offers a unified setting for weighted commutators and weighted Calder\’on-Zygmund operators, which is even interesting in the scalar case. Moreover, we will show that sharp bounds for weighted commutators imply those for Calder\’on-Zygmund operators and vice versa. This is joint work with Joshua Isralowitz, Israel Rivera-Rios, Andrei Stoica, and Sergei Treil.​