Heat kernels and Hardy spaces on non-tangentially accessible domains with applications to global regularity of inhomogeneous Dirichlet problems

Let n ≥ 2 n\ge 2 and alttext="normal upper Omega"> Ω encoding="application/x-tex">\Omega be a bounded non-tangentially accessible domain (for short, NTA domain) of alttext="double-struck R Superscript n"> mathvariant="double-struck">R encoding="application/x-tex">\mathbb {R}^n . Assume that alttext="upper L Subscript D"> L D encoding="application/x-tex">L_D is second-order divergence form elliptic operator having real-valued, bounded, measurable coefficients on squared left-parenthesis normal Omega right-parenthesis"> ( stretchy="false">) encoding="application/x-tex">L^2(\Omega ) with the Dirichlet boundary condition. The main aim this article threefold. First, authors prove heat kernels alttext="left-brace K t Super D Baseline right-brace greater-than 0"> fence="false" stretchy="false">{ K t stretchy="false">} &gt; 0 encoding="application/x-tex">\{K_t^{L_D}\}_{t&gt;0} generated by are Hölder continuous. Second, for any alttext="p element-of 0 comma 1 right-bracket"> p ∈<!-- ∈ <mml:mo>, 1 stretchy="false">] encoding="application/x-tex">p\in (0,1] , introduce ‘geometrical’ Hardy space H r p H r encoding="application/x-tex">H^p_r(\Omega restricting element double-struck n encoding="application/x-tex">H^p(\mathbb {R}^n) to show that, when StartFraction Over plus delta EndFraction + δ<!-- δ </mml:mfrac> (\frac {n}{n+\delta _0},1] right-parenthesis equals Sub = )=H^p(\Omega )=H^p_{L_D}(\Omega equivalent quasi-norms, where encoding="application/x-tex">H^p(\Omega encoding="application/x-tex">H^p_{L_D}(\Omega respectively denote associated alttext="delta encoding="application/x-tex">\delta _0\in critical index continuity Third, as applications, obtain global gradient estimates in both encoding="application/x-tex">L^p(\Omega (1,p_0) z z encoding="application/x-tex">H^p_z(\Omega {n}{n+1},1] inhomogeneous problem equations domains, 2 infinity mathvariant="normal">∞<!-- ∞ encoding="application/x-tex">p_0\in (2,\infty constant depending only alttext="n"> encoding="application/x-tex">n coefficient matrix Here, defined supported overbar"> accent="false">¯<!-- ¯ </mml:mover> encoding="application/x-tex">\overline {\Omega } denotes closure It worth pointing out range estimate scale Lebesgue spaces sharp above results established without additional assumptions