Collaborative Research: Calderon-Zygmund Operators in Highly Irregular Environments, and Applications

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Calderon-Zygmund operators are mathematical objects that play an important role in the understanding of many physical phenomena, ranging from heat transfer to turbulence in dynamical systems. The classical theory of these operators was designed to work on smooth functions. However, nature often provides us with very irregular media with which to engage. This creates the need for a very low-regularity form of the theory of singular integrals, which the principal investigators on this project have constructed. A consequence of the low-regularity theory is that through the action of Calderon-Zygmund operators on a set in a Euclidean space of a very high dimension, one can sometimes conclude that the set itself is of a much lower dimension than the ambient space, an important piece of information from the perspective of data science. To refine this approach to data analysis is one of the main goals of this project. This project considers several problems in nonhomogeneous harmonic analysis, geometric measure theory, and spectral theory. The common theme uniting the problems is the behavior of singular operators with very good (Calderon-Zygmund) kernels in very bad environments (e.g., on sets with no a priori structure, in spaces with matrix weights). Specifically, the project will pursue the following avenues of research: (1) the David-Semmes problem to characterize the rectifiability of sets and measures in high-dimensional Euclidean space in terms of the boundedness of the corresponding Riesz transforms; (2) the geometry of reflection-less measures; (3) the geometric characterization of higher-dimensional analogues of positive analytic capacity; (4) two-weight estimates for very simple singular operators in the non-Hilbert setting; and (5) sharp estimates for classical operators with matrix weights. Singular integral operators with respect to bad measures and very irregular sets appear naturally in many problems of analysis. One of the reasons for their increasing interest in recent years has been the study of analytic capacity. While the theory for the two-dimensional case (i.e., the Cauchy transform on the complex plane) and the theory of analytic capacity that emerged as its by-product are now very well understood, the analogous theory in higher dimensions has not been fully developed. The main roadblock here is the lack of geometric tools in higher dimensions. Additionally, in higher dimensions, nonhomogeneous situations arise more often than in the plane and more often one might expect. For example, boundary value problems in (otherwise smooth) domains with cusps lead to nonhomogeneous problems, because, unlike what happens in the two-dimensional setting, surface measure on the boundary of such a domain is non-doubling. This becomes an even more vexing problem if one wants to consider harmonic measure estimates for domains on whose boundaries "surface measure" is practically arbitrary. This is an important issue that the project seeks to confront.