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 \(n\ge 2\) and \(\Omega \) be a bounded Lipschitz domain of \(\mathbb {R}^n\). Assume that \(L_R\) is a second-order divergence form elliptic operator having real-valued, bounded, symmetric, and measurable coefficients on \(L^2(\Omega )\) with the Robin boundary condition. In this article, via first obtaining the Hölder estimate of the heat kernels of \(L_R\), the authors establish a new atomic characterization of the Hardy space \(H^p_{L_R}(\Omega )\) associated with \(L_R\). Using this, the authors further show that, for any given \(p\in (\frac{n}{n+\delta _0},1]\),

$$\begin{aligned} H^p_z(\Omega )+L^\infty (\Omega )=H^p_{L_N}(\Omega )=H^p_{L_R}(\Omega )\subsetneqq H^p_{L_D}(\Omega )=H^p_r(\Omega ), \end{aligned}$$

where \(H^p_{L_D}(\Omega )\) and \(H^p_{L_N}(\Omega )\) denote the Hardy spaces on \(\Omega \) associated with the corresponding elliptic operators respectively having the Dirichlet and the Neumann boundary conditions, \(H^p_z(\Omega )\) and \(H^p_r(\Omega )\) respectively denote the “supported type” and the “restricted type” Hardy spaces on \(\Omega \), and \(\delta _0\in (0,1]\) is the critical index depending on the operators \(L_D\), \(L_N\), and \(L_R\). The authors then obtain the boundedness of the Riesz transform \(\nabla L_R^{-1/2}\) on the Lebesgue space \(L^{p}(\Omega )\) when \(p\in (1,\infty )\) [if \(p>2\), some extra assumptions are needed] and its boundedness from \(H_{L_R}^{p}(\Omega )\) to \(L^{p}(\Omega )\) when \(p\in (0,1]\) or to \(H^{p}_r(\Omega )\) when \(p\in (\frac{n}{n+1},1]\). As applications, the authors further obtain the global regularity estimates, in \(L^{p}(\Omega )\) when \(p\in (0,p_0)\) and in \(H^{p}_r(\Omega )\) when \(p\in (\frac{n}{n+1},1]\), for the inhomogeneous Robin problem of \(L_R\) on \(\Omega \), where \(p_0\in (2,\infty )\) is a constant depending only on n, \(\Omega \), and the operator \(L_R\). The main novelties of these results are that the range \((0,p_0)\) of p for the global regularity estimates in the scale of \(L^p(\Omega )\) is sharp and that, in some sense, the space \(X{:}{=}H^1_{L_R}(\Omega )\) is also optimal to guarantee both the boundedness of \(\nabla L^{-1/2}_R\) from X to \(L^1(\Omega )\) or to \(H^1_r(\Omega )\) and the global regularity estimate \(\Vert \nabla u\Vert _{L^{\frac{n}{n-1}} (\Omega ;\,\mathbb {R}^n)}\le C\Vert f\Vert _{X}\) for inhomogeneous Robin problems with C being a positive constant independent of both u and f.