Problems in Function Theory and Operator Theory

摘要：申请人拟从事泛函算子理论和欧几里得调和分析中的几个相关问题的研究。他计划利用他最近与Arcozzi和Sawyer的工作作为研究Besov空间内插序列的基础。他提出用相空间方法研究伪微分算子的特征向量，以及其他极值问题的解。他还计划利用他最近与Holland合作的关于Bergman核函数的渐近行为的研究，作为研究广义西格尔-巴格曼空间是否可以用来给出平面的量子变形的起点。最后，他计划研究最近在高阶汉克尔算子研究中出现的某些技术是否在算子理论中有更广泛的应用。所提出的工作将促进对全纯函数空间的泛函算子理论和欧几里得调和分析中出现的算子的理解。它还将开发在这些领域具有普遍适用性的新观点和技术。更一般地说，所提出的工作是在算子理论，函数理论和谐波分析的接口。作品的统一主题是系统地使用相空间方法，既作为技术工具，更重要的是，作为统一的概念结构。无论是在历史上还是在最近，这些工具和观点都对理论数学中的重要新技术以及数学在科学和工程中的主要新应用产生了很大的影响。最近lacey和Thiele关于双线性算子的工作是前者的一个例子，小波及其相关方法对信号处理的影响是后者的一个引人注目的例子。当然，这些领域的进一步研究将继续在数学内外产生重要的额外贡献。

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.