Riesz transform and Hardy spaces related to elliptic operators having Robin boundary conditions on Lipschitz domains with their applications to optimal endpoint regularity estimates

Let \(n\ge 2\) $n\ge 2$ and \(\Omega \) $\mathrm{\Omega }$ be a bounded Lipschitz domain of \(\mathbb {R}^n\) ${\mathbb{R}}^n$ . Assume that \(L_R\) $L_R$ is a second-order divergence form elliptic operator having real-valued, bounded, symmetric, and measurable coefficients on \(L^2(\Omega )\) $L^2(\mathrm{\Omega })$ with the Robin boundary condition. In this article, via first obtaining the Hölder estimate of the heat kernels of \(L_R\) $L_R$ , the authors establish a new atomic characterization of the Hardy space \(H^p_{L_R}(\Omega )\) $H_{L_R}^p(\mathrm{\Omega })$ associated with \(L_R\) $L_R$ . Using this, the authors further show that, for any given \(p\in (\frac{n}{n+\delta _0},1]\) $p\in (\frac{n}{n+{\delta }_0},1]$ , \(\begin{aligned} H^p_z(\Omega )+L^\infty (\Omega )=H^p_{L_N}(\Omega )=H^p_{L_R}(\Omega )\subsetneqq H^p_{L_D}(\Omega )=H^p_r(\Omega ), \end{aligned}\) $$\begin{array}{c}\hfill H_z^p(\mathrm{\Omega })+L^{\infty }(\mathrm{\Omega })=H_{L_N}^p(\mathrm{\Omega })=H_{L_R}^p(\mathrm{\Omega })⫋H_{L_D}^p(\mathrm{\Omega })=H_r^p(\mathrm{\Omega }),\end{array}$$ where \(H^p_{L_D}(\Omega )\) $H_{L_D}^p(\mathrm{\Omega })$ and \(H^p_{L_N}(\Omega )\) $H_{L_N}^p(\mathrm{\Omega })$ denote the Hardy spaces on \(\Omega \) $\mathrm{\Omega }$ associated with the corresponding elliptic operators respectively having the Dirichlet and the Neumann boundary conditions, \(H^p_z(\Omega )\) $H_z^p(\mathrm{\Omega })$ and \(H^p_r(\Omega )\) $H_r^p(\mathrm{\Omega })$ respectively denote the “supported type” and the “restricted type” Hardy spaces on \(\Omega \) $\mathrm{\Omega }$ , and \(\delta _0\in (0,1]\) ${\delta }_0\in (0,1]$ is the critical index depending on the operators \(L_D\) $L_D$ , \(L_N\) $L_N$ , and \(L_R\) $L_R$ . The authors then obtain the boundedness of the Riesz transform \(\nabla L_R^{-1/2}\) $\mathrm{\nabla }{L}_R^{-1/2}$ on the Lebesgue space \(L^{p}(\Omega )\) $L^p(\mathrm{\Omega })$ when \(p\in (1,\infty )\) $p\in (1,\infty )$ [if \(p>2\) $p>2$ , some extra assumptions are needed] and its boundedness from \(H_{L_R}^{p}(\Omega )\) $H_{L_R}^p(\mathrm{\Omega })$ to \(L^{p}(\Omega )\) $L^p(\mathrm{\Omega })$ when \(p\in (0,1]\) $p\in (0,1]$ or to \(H^{p}_r(\Omega )\) $H_r^p(\mathrm{\Omega })$ when \(p\in (\frac{n}{n+1},1]\) $p\in (\frac{n}{n+1},1]$ . As applications, the authors further obtain the global regularity estimates, in \(L^{p}(\Omega )\) $L^p(\mathrm{\Omega })$ when \(p\in (0,p_0)\) $p\in (0,p_0)$ and in \(H^{p}_r(\Omega )\) $H_r^p(\mathrm{\Omega })$ when \(p\in (\frac{n}{n+1},1]\) $p\in (\frac{n}{n+1},1]$ , for the inhomogeneous Robin problem of \(L_R\) $L_R$ on \(\Omega \) $\mathrm{\Omega }$ , where \(p_0\in (2,\infty )\) $p_0\in (2,\infty )$ is a constant depending only on n, \(\Omega \) $\mathrm{\Omega }$ , and the operator \(L_R\) $L_R$ . The main novelties of these results are that the range \((0,p_0)\) $(0,p_0)$ of p for the global regularity estimates in the scale of \(L^p(\Omega )\) $L^p(\mathrm{\Omega })$ is sharp and that, in some sense, the space \(X{:}{=}H^1_{L_R}(\Omega )\) $X:=H_{L_R}^1(\mathrm{\Omega })$ is also optimal to guarantee both the boundedness of \(\nabla L^{-1/2}_R\) $\mathrm{\nabla }{L}_R^{-1/2}$ from X to \(L^1(\Omega )\) $L^1(\mathrm{\Omega })$ or to \(H^1_r(\Omega )\) $H_r^1(\mathrm{\Omega })$ and the global regularity estimate \(\Vert \nabla u\Vert _{L^{\frac{n}{n-1}} (\Omega ;\,\mathbb {R}^n)}\le C\Vert f\Vert _{X}\) ${\Vert \mathrm{\nabla }u\Vert }_{L^{\frac{n}{n-1}}(\mathrm{\Omega };{\mathbb{R}}^n)}\le C{\Vert f\Vert }_X$ for inhomogeneous Robin problems with C being a positive constant independent of both u and f.