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