Schedule - 2020

Presently, it is well understood what geometric features are necessary and sufficient to guarantee the boundedness of convolution-type singular integral operators (SIO's) on Lebesgue spaces. This being said, dealing with other function spaces where membership entails more than a mere size condition (like Sobolev spaces, Hardy spaces, or the John-Nirenberg space BMO) requires new techniques. In this talk I will explore recent progress in this regard, and follow up the implications of such advances into the realm of boundary value problems.

It is well known the importance of the BMO space of functions with bounded mean oscillation especially due to the famous John-Nirenberg theorem of the early 60's of the last century. This result is the archetypical self-improving result in Analysis. In this talk we will show that there is another self-improving phenomenon attached to this class of functions which roughly gives a way of defining BMO using much weaker conditions than the usual L1 oscillation. These results improved a recent work by Logunov-Slavin-Stolyarov-Vasyunin-Zatitskiy. Our method is more flexible yielding sharp results under rougher geometries.

If there is enough time I will show some self-improving phenomenon considered for first time by B. Muckenhoupt and R. Wheeden with weights which turned out to be very useful in different situations like in the extrapolation theory.

The first part is joint work with J. Canto and E. Rela and the second part with J. Canto.

Decomposition techniques such as atomic, molecular, wavelet and wave-packet expansions provide a multi-scale refinement of Fourier analysis and exploit a rather simple concept: "waves with very different frequencies are almost invisible to each other". Starting with the classical Calderón-Zygmund and Littlewood-Paley decompositions, many of these useful techniques have been developed around the study of singular integral operators. By breaking an operator or splitting the functions on which it acts into non-interacting almost orthogonal pieces, these tools capture subtle cancelations and quantify properties of an operator in terms of norm estimates in function spaces. This type of analysis has been used to study linear operators with tremendous success. More recently, similar decomposition techniques have been pushed to the analysis of new multilinear operators that arise in the study of (para) product-like operations, commutators, null-forms and other nonlinear functional expressions. In this talk we will present some of our contributions in the study of multilinear singular integrals and function spaces, and their applications to the development of Leibniz rules for fractional derivatives.

In joint work with Almut Burchard (University of Toronto) and Ryan Gibara (Université Laval), we study both the decreasing rearrangement f* and the symmetric decreasing rearrangement Sf of a rearrangeable function f of bounded mean oscillation. We give new bounds on the BMO norm of the rearrangements, and also show that the mapping taking f to f* is continuous when acting on rearrangeable functions with vanishing mean oscillation (VMO), but not on BMO.

It is well known that asymptotics of orthogonal polynomials (in some form) are used in establishing universality limits in random matrices. Moreover, the area of random matrices has done a great deal to stimulate new research in orthogonal polynomials. We show that universality limits imply "local ratio limits" for orthogonal polynomials both at the edge of the spectrum and in the bulk. Moreover, we show how uniform bounds on orthogonal polynomials follow if we know that the zeros of the orthogonal polynomials of degree n and n-1 satisfy a spacing condition, and are even equivalent to this condition. This is joint work with Eli Levin of the Open University of Israel.

We survey recent developments in the following classical inequalities for polynomials: Bernstein, Nikolskii (Reverse Hölder), Hardy--Littlewood (related to the fractional integration theorem), and Remez. We will discuss various methods to obtain these estimates from harmonic analysis, interpolation and approximation theory.