Weighted local Orlicz-Hardy spaces on domains and their applications in inhomogeneous Dirichlet and Neumann problems

Abstract:

Let $\Omega$ be either $\mathbb {R}^n$ or a strongly Lipschitz domain of $\mathbb {R}^n$, and $\omega \in A_{\infty }(\mathbb {R}^n)$ (the class of Muckenhoupt weights). Let $L$ be a second-order divergence form elliptic operator on $L^2 (\Omega )$ 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 $\mu \in (0,1]$. Let $\Phi$ be a continuous, strictly increasing, subadditive, positive and concave function on $(0,\infty )$ of critical lower type index $p_{\Phi }^-\in (0,1]$. In this paper, the authors first introduce the “geometrical” weighted local Orlicz-Hardy spaces $h^{\Phi }_{\omega , r}(\Omega )$ and $h^{\Phi }_{\omega , z}(\Omega )$ via the weighted local Orlicz-Hardy spaces $h^{\Phi }_{\omega }(\mathbb {R}^n)$, 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 $p_{\Phi }^-\in (n/(n+\mu ),1]$. Second, the authors furthermore establish three equivalent characterizations of $h^{\Phi }_{\omega , r}(\Omega )$ in terms of the grand maximal function, the radial maximal function and the atomic decomposition when the complement of $\Omega$ is unbounded and $p_{\Phi }^-\in (0,1]$. Third, as applications, the authors prove that the operators $\nabla ^2{\mathbb G}_D$ are bounded from $h^{\Phi }_{\omega , r}(\Omega )$ to the weighted Orlicz space $L^{\Phi }_{\omega }(\Omega )$, and from $h^{\Phi }_{\omega , r}(\Omega )$ to itself when $\Omega$ is a bounded semiconvex domain in $\mathbb {R}^n$ and $p_{\Phi }^-\in (\frac {n}{n+1},1]$, and the operators $\nabla ^2{\mathbb G}_N$ are bounded from $h^{\Phi }_{\omega , z}(\Omega )$ to $L^{\Phi }_{\omega }(\Omega )$, and from $h^{\Phi }_{\omega , z}(\Omega )$ to $h^{\Phi }_{\omega , r}(\Omega )$ when $\Omega$ is a bounded convex domain in $\mathbb {R}^n$ and $p_{\Phi }^-\in (\frac {n}{n+1},1]$, where ${\mathbb G}_D$ and ${\mathbb G}_N$ denote, respectively, the Dirichlet Green operator and the Neumann Green operator.