Molecular Characterizations and Dualities of Variable Exponent Hardy Spaces Associated with Operators

TL;DR

This paper introduces variable exponent Hardy spaces linked to operators with specific kernel bounds, establishing their molecular structure, duality with BMO spaces, and applications to fractional integrals and space equivalences.

Contribution

It defines and characterizes variable exponent Hardy spaces associated with operators, including molecular decomposition and duality results, advancing the understanding of these spaces.

Findings

Molecular characterization of $H_L^{p(ullet)}$ spaces.

Duality between $H_L^{p(ullet)}$ and BMO-type spaces.

Boundedness of fractional integrals on these Hardy spaces.

Abstract

Let $L$ be a linear operator on $L^2(\mathbb R^n)$ generating an analytic semigroup $\{e^{-tL}\}_{t\ge0}$ with kernels having pointwise upper bounds and $p(\cdot):\ \mathbb R^n\to(0,1]$ be a variable exponent function satisfying the globally log-H\"older continuous condition. In this article, the authors introduce the variable exponent Hardy space associated with the operator $L$, denoted by $H_L^{p(\cdot)}(\mathbb R^n)$, and the BMO-type space ${\mathrm{BMO}}_{p(\cdot),L}(\mathbb R^n)$. By means of tent spaces with variable exponents, the authors then establish the molecular characterization of $H_L^{p(\cdot)}(\mathbb R^n)$ and a duality theorem between such a Hardy space and a BMO-type space. As applications, the authors study the boundedness of the fractional integral on these Hardy spaces and the coincidence between $H_L^{p(\cdot)}(\mathbb R^n)$ and the variable exponent Hardy spaces $H^{p(\cdot)}(\mathbb R^n)$.