Sharp Sobolev embeddings and related Hardy inequalities
type: Item
instance of: scholarly article
zbMATH Open document ID: 0965.46026
publication date: 2001-04-03T00:00:00Z
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 26D15
zbMATH DE Number: 1584042
zbMATH Keywords: fractional Sobolev spaces
zbMATH Keywords: Besov spaces
zbMATH Keywords: Lorentz-Zygmund spaces
zbMATH Keywords: sharp embeddings
zbMATH Keywords: Hardy's inequality
author: D. E. Edmunds
MaRDI profile type: Publication
title: Sharp Sobolev embeddings and related Hardy inequalities
review text: The paper surveys some results on sharp Sobolev embeddings and corresponding Hardy-type inequalities in the critical case. More precisely, let \(1<p<\infty \), \(s=n/p\), and let \(A_p\) stand either for the fractional Sobolev space \(H^s_p(\mathbb R^n)\) or the Besov space \(B^s_{p,p}(\mathbb R^n)\). Suppose that \(\kappa \: (0,1] \rightarrow \mathbb R\) is positive, continuous and decreasing. The main result of the paper states that the following statements are equivalent: NEWLINENEWLINENEWLINE(i) \(\int ^1_0 (\frac {\kappa (t)f^*(t)}{1+|\log t|})^p \frac {dt}{t} \lesssim \|f|A_p\|^p \quad \text{for all} \;f\in A_p\); NEWLINENEWLINENEWLINE(ii) \(\int _{|x|<1} (\frac {\kappa (|x|)f(x)}{1+|\log |x\|})^p \frac {dx}{|x|^n} \lesssim \|f|A_p \|^p \quad \text{for all} \;f\in A_p\); NEWLINENEWLINENEWLINE(iii) \(\kappa \) is bounded.NEWLINENEWLINEFor the entire collection see [Zbl 0933.00035].
reviewed by: Bohumír Opic
description: scientific article