Estimates of Green and Martin kernels for Schrödinger operators with singular potential in Lipschitz domains
Moshe Marcus · Annales de l Institut Henri Poincaré C Analyse Non Linéaire · 2018
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)| \leq \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.