Collaborative Research: Multidimensional and Non-Homogeneous Harmonic Analysis: Bellman Functions, Perturbations of Normal Operators and Two Weight Estimates of Singular Integrals

Brown University, Providence RI

Investigators

Abstract

Proposal Numbers: 0200713 and 0200584 PIs: Alexander Volberg, Fedor Nazarov, and Serguei Treil ABSTRACT Research will be conducted on non-homogeneous harmonic analysis and on weighted norm inequalities with matrix weights. In previous work by the PIs a technique for estimating Calderon-Zygmund operators on spaces with non-doubling measures was developed, and a novel method of Bellman functions was introduced for problems in Harmonic Analysis. These techniques will be applied to solve several open problems in harmonic analysis and operator theory and to investigate new directions, which previously were deemed untractable because of the lack of technical tools. Special attention will be paid to uncovering new relations between Operator Theory, Harmonic Analysis and Stochastic Control. Among the main directions of the proposed research are: - Spectral theory for perturbations of normal operators and related problems in Harmonic Analysis: two weight estimates for Hilbert Transform and embedding theorems for the co-invariant subspaces. - Non-homogeneous T(b) theorems and their applications to generalizations of analytic capacity (electric intensity capacity); the role of curvature in higher dimensions. - Bellman function method in stochastic optimal control and in harmonic analysis; functions with matrix arguments and their applications to non-commutative problems. Harmonic analysis investigates complex processes by representing them as a sum of elementary ones (sinusoidal waves, wavelets) with well understood behavior. A central part of modern harmonic analysis deals with "singular operators" of one type or another. Such operators are pervasive in the scientific landscape: they turn up in mathematical physics, probability, engineering, image processing, etc. A new way to treat multivariate signals will be discussed. The main difficulty here is that the mathematical objects arising in such problems are non-commutative: the product depends on the order of terms, and that complicates things immensely. A new method based on Bellman functions, which originating in the stochastic optimal control, will be exploited in harmonic analysis. One important direction of research is the spectral theory for the perturbation of normal operators: results in this direction would have important consequences in mathematical physics. Another direction deals with non-commutative harmonic analysis, i.e. with treating multivariate signals.