Several estimates of Musielak--Orlicz--Hardy--Sobolev type for Schröodinger type operators

Abstract

Let $L:=-\mathrm{div}(A\nabla)+V$ be a Schrödinger type operator with the nonnegative potential $V$ belonging to the reverse H\"older class $RH_{q}(\mathbb{R}^n)$ for some $q\in(n/2,\infty]$ and $n\ge3$, where $A$ satisfies the uniformly elliptic condition. Assume that $\varphi:\,\mathbb{R}^n\times[0,\infty)\to[0,\infty)$ is a function such 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). In this article, the author proves that the operators $VL^{-1}$, $V^{1/2}\nabla L^{-1}$ and $\nabla^2L^{-1}$ are bounded from the Musielak--Orlicz--Hardy space associated with $L$, $H_{\varphi,\,L}(\mathbb{R}^n)$, to the Musielak-Orlicz space $L^{\varphi}(\mathbb{R}^n)$ or $H_{\varphi,\,L}(\mathbb{R}^n)$ under some further assumptions for $\varphi$ and $A$, which further implies a maximal inequality for $L$ in the scale of $H_{\varphi,\,L}(\mathbb{R}^n)$. All these results improve the known results by weakening the assumption for $\varphi$ and $L$.