Heat Kernels and Hardy Spaces on Non-Tangentially Accessible Domains with Applications to Global Regularity of Inhomogeneous Dirichlet Problems

Abstract

Let $n\ge 2$ and $\hat{I}\text{c◯}$ be a bounded non-tangentially accessible domain (for short, NTA domain) of ${\mathbb{R}}^n$. Assume that $L_D$ is a second-order divergence form elliptic operator having real-valued, bounded, measurable coefficients on $L^2(\hat{I}\text{c◯})$ with the Dirichlet boundary condition. The main aim of this article is threefold. First, the authors prove that the heat kernels $\{K_t^{L_D}{\}}_{t>0}$ generated by $L_D$ are Hölder continuous. Second, for any $p\in (0,1]$, the authors introduce the `geometrical' Hardy space $H_r^p(\hat{I}\text{c◯})$ by restricting any element of the Hardy space $H^p({\mathbb{R}}^n)$ to $\hat{I}\text{c◯}$, and show that, when $p\in (\frac{n}{n+\hat{I}{´}_0},1]$, $H_r^p(\hat{I}\text{c◯})=H^p(\hat{I}\text{c◯})=H_{L_D}^p(\hat{I}\text{c◯})$ with equivalent quasi-norms, where $H^p(\hat{I}\text{c◯})$ and $H_{L_D}^p(\hat{I}\text{c◯})$ respectively denote the Hardy space on $\hat{I}\text{c◯}$ and the Hardy space associated with $L_D$, and $\hat{I}{´}_0\in (0,1]$ is the critical index of the Hölder continuity for the kernels $\{K_t^{L_D}{\}}_{t>0}$. Third, as applications, the authors obtain the global gradient estimates in both $L^p(\hat{I}\text{c◯})$, with $p\in (1,p_0)$, and $H_z^p(\hat{I}\text{c◯})$, with $p\in (\frac{n}{n+1},1]$, for the inhomogeneous Dirichlet problem of second-order divergence form elliptic equations on bounded NTA domains, where $p_0\in (2,\infty )$ is a constant depending only on $n$, $\hat{I}\text{c◯}$, and the coefficient matrix of $L_D$. It is worth pointing out that the range $p\in (1,p_0)$ for the global gradient estimate in the scale of Lebesgue spaces $L^p(\hat{I}\text{c◯})$ is sharp and the above results are established without any additional assumptions on both the coefficient matrix of $L_D$, and the domain $\hat{I}\text{c◯}$.

50 pages, Submitted