Problems in Function Theory and Operator Theory

Washington University, Saint Louis MO

Investigators

Abstract

Abstract: The applicant proposes to work on several related problems in function theoretic operator theory and in Euclidean harmonic analysis. He plans to use his recent work with Arcozzi and Sawyer as a basis for the study of interpolating sequences in Besov spaces. He proposes to use phase space methods to study eigenvectors of pseudodifferential operators and as well as solutions to other extremal problems. He also plans to use his recent work with Holland on the asymptotic behavior of Bergman kernel functions as a starting point to investigate whether generalized Segal-Bargmann spaces can be used to give quantum deformations of the plane. Finally, he plans to investigate whether certain techniques that arose in recent work on higher order Hankel operators have broader application in operator theory. The work proposed will advance the understanding of the function theoretic operator theory on spaces of holomorphic functions and of operators arising in Euclidean harmonic analysis. It will also develop new viewpoints and techniques that will have general applicability in those areas. More generally, the proposed work is at the interface of operator theory, function theory, and harmonic analysis. The unifying theme of the work is the systematic use of phase space methods, both as a technical tool and, more importantly, as a unifying conceptual structure. Those tools and that viewpoint have been very productive, both historically and recently, of important new techniques in theoretical mathematics and of major new applications of mathematics to science and engineering. The recent work Lacey and Thiele on bilinear operators is an example of the first and the impact of wavelet and related methods on signal processing is a spectacular example of the second. One certainly expects that further research in these areas will continue to produce important additional contributions inside and outside of mathematics.