We investigate connections between Hardy's inequality in the whole space R-n and 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 parallel to del.parallel to(p,q) also 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 = infinity. 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 / D. Cassani, B. Ruf, C. Tarsi. - In: JOURNAL OF FUNCTIONAL ANALYSIS. - ISSN 0022-1236. - 275:12(2018 Dec 15), pp. 3303-3324.

Equivalent and attained version of Hardy's inequality in Rn

B. Ruf;C. Tarsi
2018-12-15

Abstract

We investigate connections between Hardy's inequality in the whole space R-n and 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 parallel to del.parallel to(p,q) also 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 = infinity. 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].
Hardy inequality; Sobolev-Lorentz spaces; Best constants
Settore MAT/05 - Analisi Matematica
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2434/613192
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