Variable Hardy spaces associated with Schrödinger operators on strongly Lipschitz domains with their applications to regularity for inhomogeneous Dirichlet problems
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 paper, 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.
scientific article; zbMATH DE number 7606138
scientific article