Maximal Function Characterizations of Musielak-Orlicz-Hardy Spaces Associated to Non-negative Self-adjoint Operators Satisfying Gaussian Estimates

TL;DR

This paper characterizes Musielak-Orlicz-Hardy spaces associated with certain self-adjoint operators using maximal functions, extending previous results and answering open questions in the field.

Contribution

It provides new maximal function characterizations of Musielak-Orlicz-Hardy spaces linked to non-negative self-adjoint operators with Gaussian bounds, including Schrödinger and elliptic operators.

Findings

Established maximal function characterizations of $H_{, L}(R^n)$.

Answered open questions for Schrödinger operators with reverse H"older potentials.

Extended characterizations to second-order divergence form elliptic operators.

Abstract

Let $L$ be a non-negative self-adjoint operator on $L^2(\mathbb{R}^n)$ whose heat kernels have the Gaussian upper bound estimates. Assume that the growth function $\varphi:\,\mathbb{R}^n\times[0,\infty) \to[0,\infty)$ satisfies that $\varphi(x,\cdot)$ is an Orlicz function and $\varphi(\cdot,t)\in {\mathbb A}_{\infty}(\mathbb{R}^n)$ (the class of uniformly Muckenhoupt weights). Let $H_{\varphi,\,L}(\mathbb{R}^n)$ be the Musielak-Orlicz-Hardy space introduced via the Lusin area function associated with the heat semigroup of $L$. In this article, the authors obtain several maximal function characterizations of the space $H_{\varphi,\,L}(\mathbb{R}^n)$, which, especially, answer an open question of L. Song and L. Yan under an additional mild assumption satisfied by Schr\"odinger operators on $\mathbb{R}^n$ with non-negative potentials belonging to the reverse H\"older class, and second-order divergence form elliptic operators on $\mathbb{R}^n$ with bounded measurable real coefficients.