Maximal Function Characterizations of Variable Hardy Spaces Associated with Non-negative Self-adjoint Operators Satisfying Gaussian Estimates

TL;DR

This paper characterizes variable Hardy spaces associated with certain operators using maximal functions, establishing atomic, non-tangential, and radial characterizations under Gaussian and H"older regularity conditions.

Contribution

It introduces atomic and maximal function characterizations of variable Hardy spaces linked to self-adjoint operators with Gaussian bounds, extending previous results to variable exponent settings.

Findings

Atomic characterization of $H_L^{p( ext{cdot})}(R^n)$

Non-tangential maximal function characterization

Radial maximal function characterization under H"older regularity

Abstract

Let $p(\cdot):\ \mathbb R^n\to(0,1]$ be a variable exponent function satisfying the globally $\log$-H\"older continuous condition and $L$ a non-negative self-adjoint operator on $L^2(\mathbb R^n)$ whose heat kernels satisfying the Gaussian upper bound estimates. Let $H_L^{p(\cdot)}(\mathbb R^n)$ be the variable exponent Hardy space defined via the Lusin area function associated with the heat kernels $\{e^{-t^2L}\}_{t\in (0,\infty)}$. In this article, the authors first establish the atomic characterization of $H_L^{p(\cdot)}(\mathbb R^n)$; using this, the authors then obtain its non-tangential maximal function characterization which, when $p(\cdot)$ is a constant in $(0,1]$, coincides with a recent result by Song and Yan [Adv. Math. 287 (2016), 463-484] and further induces the radial maximal function characterization of $H_L^{p(\cdot)}(\mathbb R^n)$ under an additional assumption that the heat kernels of $L$ have the H\"older regularity.