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

Let $n\ge2$ and $Ω$ 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 $L^2(Ω)$ 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 $H^p_r(Ω)$ by restricting any element of the Hardy space $H^p(\mathbb{R}^n)$ to $Ω$, and show that, when $p\in(\frac{n}{n+δ_0},1]$, $H^p_r(Ω)=H^p(Ω)=H^p_{L_D}(Ω)$ with equivalent quasi-norms, where $H^p(Ω)$ and $H^p_{L_D}(Ω)$ respectively denote the Hardy space on $Ω$ and the Hardy space associated with $L_D$, and $δ_0\in(0,1]$ 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 $L^p(Ω)$, with $p\in(1,p_0)$, and $H^p_z(Ω)$, 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 $n$, $Ω$, 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 $L^p(Ω)$ is sharp and the above results are established without any additional assumptions on both the coefficient matrix of $L_D$, and the domain $Ω$.

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