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

### Moshe Marcus

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## 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)| \leq a\limits^{¯}\:\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.

## Cite this article

Moshe Marcus, Estimates of Green and Martin kernels for Schrödinger operators with singular potential in Lipschitz domains. Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 5, pp. 1183–1200

DOI 10.1016/J.ANIHPC.2018.09.003