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Improving sharp Sobolev type inequalities by optimal remainder gradient norms

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  • We are concerned with Sobolev type inequalities in $W^{1,p}_0(\Omega )$, $\Omega \subset R^n$, 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.
    Mathematics Subject Classification: Primary: 46E35, 46E30.


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