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

TL;DR

This paper studies heat kernels and Hardy spaces on NTA domains, proving kernel regularity, characterizing Hardy spaces, and applying these results to obtain sharp global gradient estimates for elliptic PDEs with minimal assumptions.

Contribution

It establishes the Hölder continuity of heat kernels, characterizes Hardy spaces via geometric restrictions, and derives sharp global gradient estimates for elliptic equations on NTA domains without extra assumptions.

Findings

Heat kernels are Hölder continuous.

Hardy spaces are equivalent under certain conditions.

Global gradient estimates are sharp and hold under minimal assumptions.

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 $\Omega$.