Weighted local Orlicz-Hardy spaces on domains and their applications in inhomogeneous Dirichlet and Neumann problems (Q2847123)

Let \(\Omega\) be either \(\mathbb{R}^{n}\) or a strongly Lipschitz domain of \(\mathbb{R}^{n}\), and let \(\omega \in A_{\infty}(\mathbb{R}^{n})\), the class of Muckenhoupt weights. Let \(L\) be a secon-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 a suitable Gaussian property. Let \(\Phi\) be a continuous, strictly increasing, subadditive, positive, and concave function on (\(0, \infty\)).NEWLINENEWLINEThe authors introduce the weighted local Orlicz-Hardy spaces \(h_{\omega, r}^{\Phi}(\Omega)\) and \(h_{\omega, z}^{\Phi}(\Omega)\) on \(\Omega\) via the weighted local Orlicz-Hardy space \(h_{\omega}^{\Phi}(\mathbb{R}^{n})\) given in [\textit{D. Yang} and \textit{S. Yang}, Diss. Math. 478 (2011; Zbl 1241.46018)]. Namely, NEWLINE\[NEWLINEh_{\omega, r}^{\Phi}(\Omega)=\{ f \in \mathcal{D}'(\Omega):\text{ there exists }F \in h_{\omega}^{\Phi}(\mathbb{R}^{n}) \text{ such that } F|_{\Omega}=f \}NEWLINE\]NEWLINE and NEWLINE\[NEWLINEh_{\omega, z}^{\Phi}(\Omega)= \{ f \in h_{\omega, r}^{\Phi}(\mathbb{R}^{n}): f=0 \text{ on } \mathbb{R}^{n} \setminus \overline{\Omega} \}/\{ f \in h_{\omega, r}^{\Phi}(\mathbb{R}^{n}): f=0 \text{ on }\Omega \}.NEWLINE\]NEWLINENEWLINENEWLINEThe authors first obtain two equivalent characterizations of \(h_{\omega, r}^{\Phi}(\Omega)\) and \(h_{\omega, z}^{\Phi}(\Omega)\) in terms of the nontangential maximal function and the Lusin area function associated with the heat semigroup generated by \(L\).NEWLINENEWLINEThe authors furthermore establish three equivalent characterizations of \(h_{\omega, r}^{\Phi}(\Omega)\) in terms of the grand maximal function, the radial maximal function and the atomic decomposition when the complement of \(\Omega\) is unbounded.NEWLINENEWLINEAs applications, the authors prove that the operators \(\nabla^2\mathbb{G}_{D}\) are bounded from \(h_{\omega, r}^{\Phi}(\Omega)\) to the weighted Orlicz space \(L_{\omega}^{\Phi}(\Omega)\), and from \(h_{\omega, r}^{\Phi}(\Omega)\) to itself when \(\Omega\) is a bounded semiconvex domain in \(\mathbb{R}^{n}\), and the operators \(\nabla^2\mathbb{G}_{N}\) are bounded from \(h_{\omega, z}^{\Phi}(\Omega)\) to \(L_{\omega}^{\Phi}(\Omega)\), and from \(h_{\omega, z}^{\Phi}(\Omega)\) to \(h_{\omega, r}^{\Phi}(\Omega)\) when \(\Omega\) is a bounded convex domain in \(\mathbb{R}^{n}\) where \(\mathbb{G}_{D}\) and \(\mathbb{G}_{N}\) denote the Dirichlet-Green operator and Neumann-Green operator, respectively.