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

We obtain a trace Hardy inequality for the Euclidean space with a bounded cut $\Sigma\subset\mathbb R^d$, $d \ge 2$. 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 $\Sigma$. We provide its applications to the heat equation on $\mathbb R^d$ with an insulating cut at $\Sigma$ and to the Schr\"odinger operator with a $\delta'$-interaction supported on $\Sigma$. We also obtain generalizations of this trace Hardy inequality for a class of unbounded cuts.