Abstract

Variable exponent spaces are a generalization of Lebesgue spaces in which the exponent is a measurable function. Most of the research done in this topic has been situated under the context of real functions. In this work, we present two examples of variable exponent spaces of analytic functions: variable exponent Hardy spaces and variable exponent Bergman spaces. We will introduce the spaces together with some basic properties and the main techniques used in the context. We will show that in both cases, the boundedness of the evaluation functionals plays a key role in the theory. We also present a section of possible directions of research in this topic.

Highlights

  • In recent years the interest for nonstandard function spaces has risen due to several considerations including their need in some fields of applied mathematics, differential equations, and the possibility to study extensions and generalizations of classical spaces

  • A first introduction to them was due to Orlicz in 1931. He considered measurable functions u such that ð1 juðxÞjpðxÞ dx < ∞, the usual exponent p from the classical theory of Lebesgue spaces is replaced by a suitable function pðÁÞ

  • This is a simple fact that depends on the dilation-invariant nature of Bergman Spaces, something that does not hold in the case of variable exponent Bergman spaces

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Summary

Introduction

In recent years the interest for nonstandard function spaces has risen due to several considerations including their need in some fields of applied mathematics, differential equations, and the possibility to study extensions and generalizations of classical spaces. One of such type of spaces is the variable exponent Lebesgue spaces. The investigation of structural properties of variable exponent spaces and of operator theory in such spaces is of interest due to their intriguing mathematical structure— worthy of investigation—but because such spaces appear in applications.

Variable exponents
Variable exponent Hardy spaces
Mollified dilations
Carleson measures
Zero sets
Sampling sequences
Operators in variable exponent Bergman spaces
Analytic Besov spaces

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