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

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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. ***