A weighted Hardy type inequality and its applications

Abstract In this paper we prove a new Hardy type inequality and as a consequence we establish embedding results for a certain Sobolev space E 1 , p ( R + n ) defined on the upper half-space. Precisely, for 1 p n we obtain an embedding from E 1 , p ( R + n ) into weighted Lebesgue spaces. In the border-line case p = n , we derive some Trudinger-Moser type inequalities, and in the case p > n we obtain a Morrey's type inequality.