Anisotropic Variable Campanato-Type Spaces and Their Carleson Measure Characterizations

TL;DR

This paper introduces anisotropic variable Campanato-type spaces, explores their duality with anisotropic variable Hardy spaces, and establishes Carleson measure characterizations using atomic decompositions and tent spaces.

Contribution

It develops the theory of anisotropic variable Campanato-type spaces, including duality, equivalent characterizations, and Carleson measure descriptions, extending classical harmonic analysis tools.

Findings

Established duality between Campanato-type and Hardy spaces.

Provided atomic and finite atomic characterizations.

Derived Carleson measure characterizations for these spaces.

Abstract

Let $p(\cdot):\ {\mathbb{R}^n}\to(0,\infty)$ be a variable exponent function satisfying the globally log-H\"{o}lder continuous condition and $A$ a general expansive matrix on ${\mathbb{R}^n}$. In this article, the authors introduce the anisotropic variable Campanato-type spaces and give some applications. Especially, using the known atom and finite atom characterizations of anisotropic variable Hardy space $H_A^{p(\cdot)}(\mathbb{R}^n)$, the authors prove that this Campanato-type space is the appropriate dual space of $H_A^{p(\cdot)} (\mathbb{R}^n)$ with full range $p(\cdot)$. As applications, the authors first deduce several equivalent characterizations of these Campanato-type spaces. Furthermore, the authors also introduce the anisotropic variable tent spaces and show their atomic decomposition. Combining this and the obtained dual theorem, the Carleson measure characterizations of these anisotropic variable Campanato-type spaces are established.