Weighted Local Orlicz-Hardy Spaces on Domains and Their Applications in Inhomogeneous Dirichlet and Neumann Problems

Abstract

Let $Ω$ be either $\mathbb{R}^n$ or a strongly Lipschitz domain of $\mathbb{R}^n$, and $ω\in A_{\infty}(\mathbb{R}^n)$ (the class of Muckenhoupt weights). Let $L$ be a second order divergence form elliptic operator on $L^2 (Ω)$ with the Dirichlet or Neumann boundary condition, and assume that the heat semigroup generated by $L$ has the Gaussian property $(G_1)$ with the regularity of their kernels measured by $μ\in(0,1]$. Let $Φ$ be a continuous, strictly increasing, subadditive, positive and concave function on $(0,\infty)$ of critical lower type index $p_Φ^-\in(0,1]$. In this paper, the authors introduce the "geometrical" weighted local Orlicz-Hardy spaces $h^Φ_{ω,\,r}(Ω)$ and $h^Φ_{ω,\,z}(Ω)$ via the weighted local Orlicz-Hardy spaces $h^Φ_ω(\mathbb{R}^n)$, and obtain their two equivalent characterizations in terms of the nontangential maximal function and the Lusin area function associated with the heat semigroup generated by $L$ when $p_Φ^-\in(n/(n+μ),1]$. As applications, the authors prove that the operators $\nabla^2{\mathbb G}_D$ are bounded from $h^Φ_{ω,\,r}(Ω)$ to the weighted Orlicz space $L^Φ_ω(Ω)$, and from $h^Φ_{ω,\,r}(Ω)$ to itself when $Ω$ is a bounded semiconvex domain in $\mathbb{R}^n$ and $p_Φ^-\in(\frac{n}{n+1},1]$, and the operators $\nabla^2{\mathbb G}_N$ are bounded from $h^Φ_{ω,\,z}(Ω)$ to $L^Φ_ω(Ω)$, and from $h^Φ_{ω,\,z}(Ω)$ to $h^Φ_{ω,\,r}(Ω)$ when $Ω$ is a bounded convex domain in $\mathbb{R}^n$ and $p_Φ^-\in(\frac{n}{n+1},1]$, where ${\mathbb G}_D$ and ${\mathbb G}_N$ denote, respectively, the Dirichlet Green operator and the Neumann Green operator.

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