Variable Hardy spaces associated with Schrödinger operators on strongly Lipschitz domains with their applications to regularity for inhomogeneous Dirichlet problems

Abstract

Let $n\ge 3$, $\Omega$ be a strongly Lipschitz domain of ${\mathbb {R}}^n$, and $p(\cdot ):\ {\mathbb {R}}^n\rightarrow (0,1]$ a variable exponent function satisfying the globally log-Hölder continuous condition. Assume that $L_\Omega :=-\Delta +V$ is a Schrödinger operator on $L^2(\Omega )$ with the Dirichlet boundary condition, where $\Delta$ denotes the Laplace operator and the nonnegative potential V belongs to the reverse Hölder class $RH_{q_0}({\mathbb {R}}^n)$ for some $q_0\in (n/2,\infty ]$. In this article, the authors first introduce the variable Hardy space $H_{L_\Omega }^{p(\cdot )}(\Omega )$ associated with $L_\Omega$ on $\Omega$, via the Lusin area function associated with $L_\Omega$, and the “geometrical” variable Hardy space $H_{L_{{\mathbb {R}}^n},\,r}^{p(\cdot )}(\Omega )$, via the variable Hardy space $H_{L_{{\mathbb {R}}^n}}^{p(\cdot )}({\mathbb {R}}^n)$ associated with the Schrödinger operator $L_{{\mathbb {R}}^n}:=-\Delta +V$ on ${\mathbb {R}}^n$, and then prove that $H_{L_\Omega }^{p(\cdot )}(\Omega ) =H_{L_{{\mathbb {R}}^n},\,r}^{p(\cdot )}(\Omega )$ with equivalent quasi-norms. As an application, the authors show that, when $\Omega$ is a bounded, simply connected, and semiconvex domain of ${\mathbb {R}}^n$ and the nonnegative potential V belongs to the reverse Hölder class $RH_{q_0}({\mathbb {R}}^n)$ for some $q_0\in (\max \{n/2,2\},\infty ]$, the operators $VL^{-1}_\Omega$ and $\nabla ^2L^{-1}_\Omega$ are bounded from $H_{L_{{\mathbb {R}}^n},\,r}^{p(\cdot )}(\Omega )$ to the variable Lebesgue space $L^{p(\cdot )}(\Omega )$, or to itself. As a corollary, the second-order regularity for the inhomogeneous Dirichlet problems of the corresponding Schrödinger equations in the scale of variable Hardy spaces $H_{L_{{\mathbb {R}}^n},\,r}^{p(\cdot )}(\Omega )$ is obtained.