Problems in Function Theory and Operator Theory

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Abstract:The applicant proposes to work on several related problems in functiontheoretic operator theory and in Euclidean harmonic analysis. He plans touse his recent work with Arcozzi and Sawyer as a basis for the study ofinterpolating sequences in Besov spaces. He proposes to use phase spacemethods to study eigenvectors of pseudodifferential operators and as wellas solutions to other extremal problems. He also plans to use his recentwork with Holland on the asymptotic behavior of Bergman kernel functionsas a starting point to investigate whether generalized Segal-Bargmannspaces can be used to give quantum deformations of the plane. Finally, heplans to investigate whether certain techniques that arose in recent workon higher order Hankel operators have broader application in operatortheory. The work proposed will advance the understanding of the functiontheoretic operator theory on spaces of holomorphic functions and ofoperators arising in Euclidean harmonic analysis. It will also develop newviewpoints and techniques that will have general applicability in thoseareas.More generally, the proposed work is at the interface of operator theory,function theory, and harmonic analysis. The unifying theme of the work isthe systematic use of phase space methods, both as a technical tool and,more importantly, as a unifying conceptual structure. Those tools andthat viewpoint have been very productive, both historically and recently,of important new techniques in theoretical mathematics and of major newapplications of mathematics to science and engineering. The recent workLacey and Thiele on bilinear operators is an example of the first and theimpact of wavelet and related methods on signal processing is aspectacular example of the second. One certainly expects that furtherresearch in these areas will continue to produce important additionalcontributions inside and outside of mathematics.