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 $p=n\geq2.$ In this paper we provide an optimal Sobolev-type improvement of this substitute, analogous to the corresponding improvements obtained for $p=2<n$ in S. Filippas, A. Tertikas, Optimizing improved Hardy inequalities, J. Funct. Anal. 192 (1) (2002) 186--233, and for $p>n\geq1$ in G. Psaradakis, An optimal Hardy-Morrey inequality, Calc. Var. Partial Differential Equations 45 (3-4) (2012) 421--441.