A Leray–Trudinger inequality

We consider a multidimensional version of an inequality due to Leray as a substitute for Hardy's inequality in the case p=n≥2p=n≥2. In this paper we provide an optimal Sobolev-type improvement of this substitute, analogous to the corresponding improvements obtained for p=2n≥1p>n≥1 in G. Psaradakis (2012) [26].