# Carleson measure spaces with variable exponents and their applications

## TL;DR

This paper introduces variable exponent Carleson measure spaces and demonstrates their duality with variable Hardy spaces, establishing equivalences with other variable exponent function spaces and proving boundedness of certain operators.

## Contribution

It develops the theory of Carleson measure spaces with variable exponents, including duality, space equivalences, and operator boundedness, extending classical analysis to variable exponent settings.

## Key findings

- Dual space of $H^{p( ext{·})}$ is $CMO^{p( ext{·})}$.
- $CMO^{p( ext{·})}$ coincides with Campanato and Hölder-Zygmund spaces with variable exponents.
- Boundedness of Calderón-Zygmund operators on $CMO^{p( ext{·})}$.

## Abstract

In this paper, we introduce the Carleson measure spaces with variable exponents $CMO^{p(\cdot)}$. By using discrete Littlewood$-$Paley$-$Stein analysis as well as Frazier and Jawerth's $\varphi-$transform in the variable exponent settings, we show that the dual space of the variable Hardy space $H^{p(\cdot)}$ is $CMO^{p(\cdot)}$. As applications, we obtain that Carleson measure spaces with variable exponents $CMO^{p(\cdot)}$, Campanato space with variable exponent $\mathfrak{L}_{q,p(\cdot),d}$ and H\"older-Zygmund spaces with variable exponents $\mathcal {\dot{H}}_d^{p(\cdot)}$ coincide as sets and the corresponding norms are equivalent. Via using an argument of weak density property, we also prove the boundedness of Calder\'{o}n-Zygmund singular integral operator acting on $CMO^{p(\cdot)}$.