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

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 $\ddot{I}†:{\mathbb{R}}^n\times [0,\infty )\to [0,\infty )$ satisfies that $\ddot{I}†(x,\cdot )$ is an Orlicz function and $\ddot{I}†(\cdot ,t)\in {\mathbb{A}}_{\infty }({\mathbb{R}}^n)$ (the class of uniformly Muckenhoupt weights). Let $H_{\ddot{I}†,L}({\mathbb{R}}^n)$ 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 $H_{\ddot{I}†,L}({\mathbb{R}}^n)$, which, especially, answer an open question of L. Song and L. Yan under an additional mild assumption satisfied by Schrödinger operators on ${\mathbb{R}}^n$ with non-negative potentials belonging to the reverse Hölder class, and second-order divergence form elliptic operators on ${\mathbb{R}}^n$ with bounded measurable real coefficients.

28 pages; Submitted