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

Original · EN

Consider operators of the form Lγᵛ:=Δ+γV in a bounded Lipschitz domain Ω⊂ Rⁿ. Assume that V∈ C¹(Ω) satisfies |V(x)| ≤ a distance(x,∂Ω)⁻² for every x∈ Ω and γ is a number in a range (γ-,γ+) described in the introduction. The model case is V(x)= distance(x,F)⁻² where F is a closed subset of ∂Ω and γ< cₕ(V)= Hardy constant for V. We provide sharp two sided estimates of the Green and Martin kernel for Lγᵛ in Ω. In addition we establish a pointwise version of the 3G inequality.