Collaborative research: Weighted Estimates with Matrix Weights and Non-Homogeneous Harmonic Analysis

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Calderon-Zygmund operators are objects that are largely responsible for our understanding of a number of physical phenomena, from heat transfer to turbulence. Recently, these operators have found application in big data analysis. The classical theory was built by Alberto Calderon and Antoni Zygmund in the early 1950s, and was intrinsically designed to work on smooth objects. However, nature often puzzles us with very irregular medium. Thus, the need arose for a very low regularity Calderon-Zygmund theory, which the three PIs have, in fact, constructed. One possible application of such low regularity theory is that by the action of Calderon-Zygmund operators on a set in a space of a very high dimension, we can conclude that the set itself is nicely structured and can be analyzed. This is a typical big data problem. Our other recent observation is that well-studied problems for such an operator can be dualized to provide new information for analysis on the hypercube - another widely used model of big data.This project will consider the following problems, presented here in their simplest by formulation: 1) Sharp end-point weak weighted estimates for the square function operator in the homogeneous setting; 2) What goes wrong in the non-homogeneous case; 3) Matrix A_2 problems for Calderon--Zygmund operators, the square function operator, their sparse operators analogies and their sharp estimates; 4) How to get from end-point estimates of the square function to geometric inequalities on the hypercube and in Gaussian space; 5) The David-Semmes problem for co-dimension higher than one; 6) Harmonic measure estimates on sets of co-dimension bigger than one; and 7) Estimates from below of singular Riesz transforms by positive geometric quantities.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.