Mathematical Sciences: Imbedding and Multiplier Theorems forTriebel-Lizorkin Spaces and Spaces of Holomorphic Functions

Verbitsky 9401493该奖项支持对定义在多个实变量或复变量的域上的函数空间相关问题的数学研究。在过去的十年中发展起来的空间在Calderon-Zygmund和偏微分算子的分析中起着重要的作用，被称为Triebel-Lizorkin空间。它们是经典的Sobolev和Besov空间的改进，因为它们在分布的小波型分解中起着关键作用而发展起来。这项工作的基础是n维复空间中加权范数不等式、Carleson测度、零光滑Besov空间和球单位的Hardy-Sobolev空间之间的一组联系。它的目标是确定离散的Triebel-Lizorkin空间在具有p次序列范数的加权序列空间中的正向嵌入和反向嵌入。此外，还将在离散极大算子和希尔伯特变换保持上述空间的直线上刻画正测度。最后，将继续就球代数的子集的容量方面与峰值内插集有关的问题进行研究。对经典函数空间的研究从关注其结构扩展到应用到对基本算子的更适用的分析的理解。随着过去二十年的发展，空间从微分方程、奇异算子和小波中获得了推动力，而不是泛函分析。这导致了一种新的、充满活力的数学分析的混合，这一提议就是例证。***

Verbitsky 9401493 This award supports mathematical research on problems related to function spaces defined on domains in several real or complex variables. The spaces, developed over the past decade play a major role in the analysis of Calderon-Zygmund and partial differential operators, are known as Triebel-Lizorkin spaces. They are refinements of the classical Sobolev and Besov spaces, developed because of the crucial role they play in wavelet-type decompositions of distributions. The basis for this work is a set of connections between weighted norm inequalities, Carleson measures, Besov spaces with zero smoothness and Hardy-Sobolev spaces in the unit of ball in complex n-dimensional space. It goals are to determine forward and reverse embedding of discrete Triebel-Lizorkin spaces into weighted sequence spaces with p-th poser norms. In addition, work will be done characterizing positive measures on the line for which the discrete maximal operator and Hilbert transform preserve the above spaces. Finally, work will continue on questions related to peak interpolation sets for the ball algebra in terms of the capacity of its subsets. Studies of the classical function spaces expanded from concerns about their structure to the application to the understanding of basic operators of more applicable analysis. As the past two decades progressed, the spaces gained their impetus from differential equations, singular operators and wavelets rather than from functional analysis. This has led to a new and vigorous blend of mathematical analysis exemplified by this proposal. ***