Hardy inequalities with optimal constants and remainder terms

Gazzola, Filippo; Grunau, Hans-Christoph; Mitidieri, Enzo

doi:10.1090/S0002-9947-03-03395-6

Hardy inequalities with optimal constants and remainder terms
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by Filippo Gazzola, Hans-Christoph Grunau and Enzo Mitidieri

Trans. Amer. Math. Soc. 356 (2004), 2149-2168

DOI: https://doi.org/10.1090/S0002-9947-03-03395-6

Published electronically: December 9, 2003

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Abstract:

We show that the classical Hardy inequalities with optimal constants in the Sobolev spaces $W_0^{1,p}$ and in higher-order Sobolev spaces on a bounded domain $\Omega \subset \mathbb {R}^n$ can be refined by adding remainder terms which involve $L^p$ norms. In the higher-order case further $L^p$ norms with lower-order singular weights arise. The case $1<p<2$ being more involved requires a different technique and is developed only in the space $W_0^{1,p}$.

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