Heat and Martin Kernel estimates for Schrödinger operators with critical Hardy potentials

Abstract Let $$\Omega $$ Ω be a bounded domain in $${{\mathbb {R}}}^N$$ R N with $$C^2$$ C 2 boundary and let $$K\subset \partial \Omega K ⊂ ∂ either submanifold of the codimension $$k<N$$ k < or point. In this article we study various problems related to Schrödinger operator $$L_{\mu } =-\Delta - \mu d_K^{-2}$$ L μ = - Δ d where $$d_K$$ denotes distance K $$\mu \le k^2/4$$ ≤ / 4 . We establish parabolic Harnack inequalities as well two-sided heat kernel Green function estimates. construct associated Martin prove existence uniqueness for corresponding value problem data given by measures. To our results introduce among other things suitable notion trace. This trace is different from one used Marcus Nguyen (Math Ann 374(1–2):361–394, 2019) thus allowing us cover whole range