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

该奖项支持在几个实数或复数变量域中定义的函数空间相关问题的数学研究。在过去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作用，被称为triiebel - lizorkin空间。它们是经典Sobolev和Besov空间的改进，因为它们在分布的小波型分解中起着至关重要的作用。本工作的基础是复n维空间中球单位中的加权范数不等式、Carleson测度、零光滑的Besov空间和Hardy-Sobolev空间之间的一组联系。它的目标是确定离散triiebel - lizorkin空间的正向和反向嵌入到具有p阶范数的加权序列空间中。此外，我们将对离散极大算子和希尔伯特变换保持上述空间的直线上的正测度进行刻画。最后，根据球代数子集的容量，将继续研究球代数的峰值插值集的相关问题。对经典函数空间的研究从关注其结构扩展到应用，再到对基本算子的理解和更适用的分析。随着过去二十年的发展，微分方程、奇异算子和小波而不是泛函分析推动了这一领域的发展。这导致了一种新的、有力的数学分析的融合，这一建议就是例证。＊＊＊

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. ***