Estimates of Green and Martin kernels for Schrödinger operators with singular potential in Lipschitz domains

Abstract

Consider operators of the form $L^{\gamma V}:=\mathrm{\Delta }+\gamma V$ in a bounded Lipschitz domain $\mathrm{\Omega }\subset {\mathbb{R}}^N$. Assume that $V\in C^1(\mathrm{\Omega })$ satisfies $|V(x)|\le \bar{a}\mathrm{dist}(x,\partial \mathrm{\Omega }{)}^{-2}$ for every $x\in \mathrm{\Omega }$ and γ is a number in a range $({\gamma }_{-},{\gamma }_{+})$ described in the introduction. The model case is $V(x)=\mathrm{dist}(x,F{)}^{-2}$ where F is a closed subset of ∂Ω and $\gamma <c_H(V)=$ Hardy constant for V. We provide sharp two sided estimates of the Green and Martin kernel for $L^{\gamma V}$ in Ω. In addition we establish a pointwise version of the 3G inequality.