Non-Homogeneous Harmonic Analysis, two weight estimates, and spectral problems

Michigan State University, East Lansing MI

Investigators

Abstract

ABSTRACT. PI's propose to concentrate their efforts on several classical problems in Analysis and Spectral Theory that remained unsolved for the last 20--50 years, due to the lack of appropriate technical tools. Among the problems are: * bilipschitz equivalence for higher dimensional analogues of analytic capacity; * two weight estimates for the Hilbert Transform; * well-posedness of the inverse scattering problem for the discrete Schrodinger operator, i.e., uniqueness of the inverse nonlinear Fourier transform; * selected problems of noncommutative harmonic analysis Although the problems span several different areas of analysis and mathematical physics, our recent research revealed striking connections between the proposed problems. To put it briefly, they all are unified by the fact that in all of them the same type of singular kernels (usually the Cauchy kernel) appears. Also, the problems share the same difficulty, the kernel got "spoiled'' by multiplication by virtually arbitrary functions (weights). Recent developments in the non-homogeneous harmonic analysis, which treats exactly this type of situations, made successful solution of the proposed problems plausible. 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 integral 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. While the theory of singular integral operators is now well developed (starting with works of Calderon and Zygmund and continued by numerous researchers after them), it deals with the operators defined on a nice "smooth" set, like the usual Euclidean space. However, in many problems one needs to investigate such operators on a "bad" set, like surfaces with singularities and even on more pathological sets. The non-homogeneous harmonic analysis was introduced by the PI's to deal exactly with such situations: recent solution by X. Tolsa of the famous subbaditivity problem for the analytic capacity is one of the most impressive applications of this PI's theory of nonhomogeneous analysis. PI's propose to attack several classical problems, where the framework of the non-homogeneous harmonic analysis appear naturally.