We investigate connections between Hardy's inequality in the whole space Rnand embedding inequalities for Sobolev–Lorentz spaces. In particular, we complete previous results due to [1, Alvino] and [28, Talenti] by establishing optimal embedding inequalities for the Sobolev–Lorentz quasinorm ‖∇⋅‖p,qalso in the range p<q, which remained essentially open since [1]. Attainability of the best embedding constants is also studied, as well as the limiting case when q=∞. Here, we surprisingly discover that the Hardy inequality is equivalent to the corresponding Sobolev–Marcinkiewicz embedding inequality. Moreover, the latter turns out to be attained by the so-called “ghost” extremal functions of [6, Brezis–Vazquez], in striking contrast with the Hardy inequality, which is never attained. In this sense, our functional approach seems to be more natural than the classical Sobolev setting, answering a question raised in [6].

Equivalent and attained version of Hardy's inequality in Rn

Cassani, D.
;
2018-01-01

Abstract

We investigate connections between Hardy's inequality in the whole space Rnand embedding inequalities for Sobolev–Lorentz spaces. In particular, we complete previous results due to [1, Alvino] and [28, Talenti] by establishing optimal embedding inequalities for the Sobolev–Lorentz quasinorm ‖∇⋅‖p,qalso in the range p
2018
http://www.elsevier.com/inca/publications/store/6/2/2/8/7/9/index.htt
Best constants; Hardy inequality; Sobolev–Lorentz spaces; Analysis
Cassani, D.; Ruf, B.; Tarsi, C.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11383/2074872
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