Trace Hardy inequality for the Euclidean space with a cut and its applications

Abstract We obtain a trace Hardy inequality for the Euclidean space with a bounded cut Σ ⊂ R d , d ≥ 2 . In this novel geometric setting, the Hardy-type inequality non-typically holds also for d = 2 . The respective Hardy weight is given in terms of the geodesic distance to the boundary of Σ. We provide its applications to the heat equation on R d with an insulating cut at Σ and to the Schrodinger operator with a δ ′ -interaction supported on Σ. We also obtain generalizations of this trace Hardy inequality for a class of unbounded cuts.