Improving sharp Sobolev type inequalities by optimal remainder gradient norms

We are concerned with Sobolev type inequalities in $W^{1,p}_0(\Omega )$, $\Omega \subset \rn$, with optimal target norms and sharp constants. Admissible remainder terms depending on the gradient are characterized. As a consequence, the strongest possible remainder norm of the gradient is exhibited. Both the case when $p< n$ and the borderline case when $p=n$ are considered. Related Hardy inequalities with remainders are also derived.