Sharp trace Hardy–Sobolev inequalities and fractional Hardy–Sobolev inequalities

Abstract

In this work we establish sharp weighted trace Hardy inequalities with trace remainder terms involving the critical Sobolev exponent corrected by a singular logarithmic weight. We show that this weight is optimal in the sense that the inequality fails for more singular weights. Then we apply these results to derive sharp Hardy inequalities and relative improvements associated with fractional s-th powers of the Laplacian, s∈(0,1). In particular, we deal with two different operators of this type, defined on bounded domains. It follows that Hardy inequalities associated with two different fractional Laplacians share the same best constant as well as they can be both sharpened by adding Sobolev type remainder term involving the same optimal logarithmic correction. Hardy type remainder terms are also considered. Our results are in direct accordance with earlier results for their non-fractional counterpart where s=1, that is the standard Laplacian.