Product-Type Operators on the Space of Fractional Cauchy Transforms

Zeng Fan,Xin Guo,Humberto Rafeiro

doi:10.1155/2022/2644844

Abstract

The space of fractional Cauchy transforms plays a central role in classical complex analysis, harmonic analysis, and geometric measure theory. In this paper, we study the boundedness and compactness of product-type operators from the space of fractional Cauchy transforms to the Zygmund-type space in terms of the function theoretic characterization of Julia–Carathéodory type.

Highlights

Let α > 0 be a real number and M be the space of all complex Borel measures on T endowed with the total variation norm. e family Fα of fractional Cauchy transforms is the collection of holomorphic functions f in D for which dμ(ξ) f(z) 􏽚
Where the infimum extends over all measures μ. e fractional Cauchy transforms space Fα plays a central role in classical complex analysis, harmonic analysis, and geometric measure theory which has phenomenal development in connection with the Calderon–Zygmundtype singular integral theory. e space Fα may be identified with M/H10, the quotient of the Banach space M by H10, and the subspace of L1 consisting of functions with mean value 0 whose conjugate belongs to the Hardy space H1
According to the Lebesgue decomposition theorem, each μ ∈ M can be written as μ μa⊕μs, where μa is absolutely continuous with respect to the Lebesgue measure and μs is singular with respect to the Lebesgue measure

Summary

Introduction

Let S S(D) be the class of all holomorphic self-maps of the unit disk D of the complex plane C, T be the boundary of D, N0 be the set of all nonnegative integers, and N be the set of all positive integers. Product-type operators on some spaces of holomorphic functions on the unit disk have become a subject of increasing interest (see [17–19] and references therein). Liu and Yu [21] investigated the boundedness and compactness of the operator DCφ from H∞ and Bloch spaces to Zygmund spaces. Zhu [23] studied the boundedness and compactness of linear operators which are obtained by taking products of multiplication, composition, and differentiation operators from Bergman-type spaces to Berstype spaces.

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Conclusion