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

Abstract

Let $n\ge2$ and $\Omega$ 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 $L^2(\Omega)$ 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 H\"older continuous. Second, for any $p\in(0,1]$, the authors introduce the `geometrical' Hardy space $H^p_r(\Omega)$ by restricting any element of the Hardy space $H^p(\mathbb{R}^n)$ to $\Omega$, and show that, when $p\in(\frac{n}{n+\delta_0},1]$, $H^p_r(\Omega)=H^p(\Omega)=H^p_{L_D}(\Omega)$ with equivalent quasi-norms, where $H^p(\Omega)$ and $H^p_{L_D}(\Omega)$ respectively denote the Hardy space on $\Omega$ and the Hardy space associated with $L_D$, and $\delta_0\in(0,1]$ is the critical index of the H\"older continuity for the kernels $\{K_t^{L_D}\}_{t>0}$. Third, as applications, the authors obtain the global gradient estimates in both $L^p(\Omega)$, with $p\in(1,p_0)$, and $H^p_z(\Omega)$, 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 $n$, $\Omega$, 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 $L^p(\Omega)$ is sharp and the above results are established without any additional assumptions on both the coefficient matrix of $L_D$, and the domain $\Omega$.