Abstract
In this note we exploit nonlinear capacity estimates in the spirit of Mitidieri-Pohozaev [ 15 ] in the context of Lorentz spaces. This from one side yields a simple proof, though non-optimal, of non-attainability of Hardy's inequality in \begin{document}$\mathbb{R}^N$\end{document} , on the other side gives a partial positive answer to a conjecture raised in [ 15 ].
Highlights
For smooth compactly supported functions in Ω ⊆ RN the following well-known Hardy inequality holds |∇u|p dx ≥ Ω N − p p |u|p p Ω |x|p dx (H )where 1 < p < N and the constant Hp = ((N − p)/p)p is the best possible for any sufficiently smooth domain containing the origin
Besides heuristic explanations of this phenomenon motivated by the pioneering work of [3, Brezis-Vazquez], see [14, Maz’ya], by which the candidate extremal functions fail to belong to the ‘natural’ Sobolev space W 1,p(Ω), only recently a deeper understanding has been reached through the study carried out in [9, Devyver-Fraas-Pinchover]
By combining the fact that the Lorentz quasi-norm in the Sobolev critical case is equal to the Hardy weighted Lp norm, with nonlinear capacity estimates, we prove that the Hardy constant is never attained when Ω = Rn, even in the very weak sense (1), provided some additional condition on N, p is assumed
Summary
For smooth compactly supported functions in Ω ⊆ RN the following well-known Hardy inequality holds It is a well known fact that the best constant in the Hardy inequality is never attained. Ruf-Tarsi], inspired by the original intuition of [3], the authors prove an equivalent version of Hardy’s inequality which is, surprisingly, attained by explicit extremal
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