Dyadic Harmonic Analysis
Ma. Cristina Pereyra
Abstract: In 2000 I gave a series of lectures in Cuernavaca on Dyadic Harmonic Analysis. The thesis was that there is a parallel dyadic world to classical harmonic analysis where often the objects and theorems are easier to grasp. Since then a lot of striking results have been produced including Hytönen’s proof of the A_2 conjecture (2012), Lacey ‘s (2014) et al proof of the two-weight boundedness of the Hilbert transform, and Lerner’s introduction in 2013 of the powerful idea, later refined by Hytönen, Lacey, and Lerner himself, that many operators can be controlled (pointwise!) by sparse positive dyadic operators. This is a very active area of research. These dyadic techniques are well-known by harmonic analysts but they have been stretched to cover settings as diverse as graphs, fractals, compact Lie groups, smooth manifolds with doubling measure, Carnot-Caratheodory spaces, etc. The link between all these seemingly disconnected setting is that they can be viewed as spaces of homogeneous type, and one can associate to such spaces multiple dyadic structures, Haar bases, and even wavelets with Hölder regularity. In these lectures I want to give you a glimpse into this world starting in the humble setting of the real line.
Harmonic Analysis on Fractals
Ricardo A. Sáenz
Abstract: In this course we define the Laplacian on a post-critically finite set, and study the basic properties of harmonic functions and spectral decomposition. We also study Poisson integrals and function spaces related to them.
Universidad de Colima
March 13 – 17, 2017
Dyadic Harmonic Analysis, Ma. Cristina Pereyra
Coloquio de Física y Matemáticas
Averaging and (Harmonic) Analysis, Ma. Cristina Pereyra
Abstract: Averages are everywhere, they help us understand all sort of data. There are several possible meaningful averages, we will review some of these and how they compare to each other, first for finite sets of numbers then for functions. In doing so we will revisit some important and useful inequalities. Averaging or smoothing is at the heart of harmonic analysis, I will try to briefly describe a few classical and not so classical instances of this connection.
Dyadic Harmonic Analysis and Weighted Inequalities, Ma. Cristina Pereyra
Abstract: In this talk we I will give you a tour of one and two-weight inequalities with emphasis on the dyadic theory and how it influences the classical Calderón-Zygmund theory using the Hilbert transform as a toy model.
University of New Mexico
April 17 – 21, 2017
Harmonic Analysis on Fractals, Ricardo A. Sáenz
Department of Mathematics and Statistics Colloquium