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

Abstract

Let $\hat{I}\text{c◯}$ be either ${\mathbb{R}}^n$ or a strongly Lipschitz domain of ${\mathbb{R}}^n$, and $\ddot{I}‰\in A_{\infty }({\mathbb{R}}^n)$ (the class of Muckenhoupt weights). Let $L$ be a second order divergence form elliptic operator on $L^2(\hat{I}\text{c◯})$ 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 $\hat{I}¼\in (0,1]$. Let $\hat{I}¦$ be a continuous, strictly increasing, subadditive, positive and concave function on $(0,\infty )$ of critical lower type index $p_{\hat{I}}{¦}^{-}\in (0,1]$. In this paper, the authors introduce the "geometrical" weighted local Orlicz-Hardy spaces $h^{\hat{I}}{¦}_{\ddot{I}‰,r}(\hat{I}\text{c◯})$ and $h^{\hat{I}}{¦}_{\ddot{I}‰,z}(\hat{I}\text{c◯})$ via the weighted local Orlicz-Hardy spaces $h^{\hat{I}}{¦}_{\ddot{I}}‰({\mathbb{R}}^n)$, 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 $p_{\hat{I}}{¦}^{-}\in (n/(n+\hat{I}¼),1]$. As applications, the authors prove that the operators ${\nabla }^2{\mathbb{G}}_D$ are bounded from $h^{\hat{I}}{¦}_{\ddot{I}‰,r}(\hat{I}\text{c◯})$ to the weighted Orlicz space $L^{\hat{I}}{¦}_{\ddot{I}}‰(\hat{I}\text{c◯})$, and from $h^{\hat{I}}{¦}_{\ddot{I}‰,r}(\hat{I}\text{c◯})$ to itself when $\hat{I}\text{c◯}$ is a bounded semiconvex domain in ${\mathbb{R}}^n$ and $p_{\hat{I}}{¦}^{-}\in (\frac{n}{n+1},1]$, and the operators ${\nabla }^2{\mathbb{G}}_N$ are bounded from $h^{\hat{I}}{¦}_{\ddot{I}‰,z}(\hat{I}\text{c◯})$ to $L^{\hat{I}}{¦}_{\ddot{I}}‰(\hat{I}\text{c◯})$, and from $h^{\hat{I}}{¦}_{\ddot{I}‰,z}(\hat{I}\text{c◯})$ to $h^{\hat{I}}{¦}_{\ddot{I}‰,r}(\hat{I}\text{c◯})$ when $\hat{I}\text{c◯}$ is a bounded convex domain in ${\mathbb{R}}^n$ and $p_{\hat{I}}{¦}^{-}\in (\frac{n}{n+1},1]$, where ${\mathbb{G}}_D$ and ${\mathbb{G}}_N$ denote, respectively, the Dirichlet Green operator and the Neumann Green operator.

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