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.
Titolo: | Equivalent and attained version of Hardy's inequality in Rn |
Autori: | |
Parole Chiave: | Hardy inequality; Sobolev-Lorentz spaces; Best constants |
Settore Scientifico Disciplinare: | Settore MAT/05 - Analisi Matematica |
Data di pubblicazione: | 15-dic-2018 |
Rivista: | |
Tipologia: | Article (author) |
Digital Object Identifier (DOI): | http://dx.doi.org/10.1016/j.jfa.2018.09.008 |
Appare nelle tipologie: | 01 - Articolo su periodico |
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