Singular integrals on curves, the Beurling-Ahlfors transform, and commutators

Washington University, Saint Louis MO

Investigators

Abstract

This project addresses properties of singular integrals, an important tool in modern analysis and its applications. Such integrals appear systematically in the context of harmonic analysis, a field of mathematics which studies properties of functions via their decomposition into components according to a suitable notion of vibrational frequency. The study of singular integrals is characterized by delicate cancellations which occur between positive and negative contributions. Some key questions to be considered in this project involve commutators. A commutator is the difference A*B-B*A of two products taken in the opposite order. The commutation property A*B = B*A is rare in general, and the size of the commutator gives a measure for the degree of non-commutativity. Classical examples are related to the uncertainty principle in quantum mechanics. The project will advance state of the art commutator theory involving singular integrals. This includes problems on curves with a geometric flavor and higher parameter variants of recent key estimates with applications to partial differential equations. Part of the project deals with an important singular integral, the Beurling–Ahlfors transform, and particularly with questions related to weighted estimates. Weighted estimates are generally desirable due to their wide use throughout harmonic analysis and its applications. Methodologically, discrete methods involving probability have recently become central in related problems, and such methods will be further developed in combination with new geometric constructions. In addition, the Principal Investigator will supervise graduate students on topics related to the proposed research, and will organize a research workshop for early-career researchers and a separate online conference for undergraduate research. This project revolves around the study of singular integrals, harmonic analysis, and commutators. Among the topics to be considered are weighted estimates for higher powers of the Beurling–Ahlfors transform, an important exemplar of the class of singular integrals. The relevant theory has recently been advanced by the discovery of counterexamples showing the failure of the famous A2 conjecture in this context, but the precise bounds are not completely understood. The investigator will also advance the theory of commutators by studying singular integrals on curves and bi-commutator analogues of recent commutator characterizations. Such commutator problems are also related to weak variants of sparse domination in the multi-parameter setting. Sparse domination methods have been central in the past decade for related problems but up to now have remained a one-parameter tool. On the methodological side, the project will further develop the method of approximate weak factorizations, which has been responsible for many recent breakthroughs, and will combine that method with other novel tools. The project will utilize dyadic-probabilistic analysis, fine-tuned counterexamples, and geometric factorizations to develop applicable harmonic analysis methods suitable for these and other questions. 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.