Hardy–Sobolev inequalities in unbounded domains and heat kernel estimates

We deal with domains with infinite inner radius. More precisely, we introduce a new geometric assumption on an exterior domain Ω⊂Rn; n⩾3 (i.e. complement of smooth compact domain not containing the origin). Under this assumption, we prove the Hardy inequality with optimal constant involving the distance to the boundary. In addition, in the case n⩾4, we improve this inequality by adding a critical Sobolev norm. Furthermore, we investigate the singular case n=3 and we show that, under some additional geometric assumption on Ω, the Hardy inequality can be improved by adding a Sobolev type term with critical exponent. Also, we prove some Hardy–Sobolev type inequalities without any geometric assumptions on Ω, which are of independent interest. Finally, we prove Harnack inequality up to the boundary for the positive solutions of the problem and we prove heat kernel estimates for small times.