Littlewood-Paley Characterization for Musielak-Orlicz-Hardy Spaces Associated with Operators

TL;DR

This paper introduces a new characterization of Musielak-Orlicz-Hardy spaces linked to a specific operator on a space of homogeneous type, using Lusin area and Littlewood-Paley functions.

Contribution

It provides a novel Lusin area function characterization for Musielak-Orlicz-Hardy spaces associated with operators on spaces of homogeneous type.

Findings

Established equivalence of Musielak-Orlicz-Hardy spaces via Lusin area and Littlewood-Paley functions.

Proved the norms of these characterizations are equivalent.

Extended the theory to spaces with Gaussian upper bounds for the semigroup kernel.

Abstract

Let $X$ be a space of homogeneous type. Assume that $L$ is an non-negative second-order self-adjoint operator on $L^2\left(X\right)$ with (heart) kernel associated to the semigroup $e^{ - tL}$ that satisfies the Gaussian upper bound. In this paper, the authors introduce a new characterization of the Musielak-Orlicz-Hardy Space $H_{\varphi, L}\left(X\right)$ associated with $L$ in terms of the Lusin area function where $\varphi$ is a growth function. Further, the authors prove that the Musielak-Orlicz-Hardy Space $H_{L,G,\varphi}\left(X\right)$ associated with $L$ in terms of the Littlewood-Paley function is coincide with $H_{\varphi, L}\left(X\right)$ and their norms are equivalent.