Remarks on inequalities of Hardy-Sobolev Type

Xiao

Description

We obtain the sharp constants of some Hardy-Sobolev-type inequalities proved by Balinsky et al. (Banach J Math Anal 2(2):94-106). 2000 Mathematics Subject Classification : Primary 26D10; 46E35. We obtain the sharp constants of some Hardy-Sobolev-type inequalities proved by Balinsky et al. (Banach J Math Anal 2(2):94-106). 2000 Mathematics Subject Classification : Primary 26D10; 46E35.

Sujets

• Hardy's inequality
• Sobolev inequality

Informations

• Publié par: biomed
• Publié le: 01 janvier 2011
• Nombre de lectures: 5
• Langue: English

Extrait

XiaoJournal of Inequalities and Applications2011,2011:132 http://www.journalofinequalitiesandapplications.com/content/2011/1/132

R E S E A R C H

Remarks

YingXiong Xiao

on

Correspondence: yxxiao. [e-mail] School of Mathematics and Statistics, Xiaogan University, Xiaogan 432000 Hubei, People’s Republic of China

inequalities

of

HardySobolev

Open Access

Type

Abstract We obtain the sharp constants of some HardySobolevtype inequalities proved by Balinsky et al. (Banach J Math Anal 2(2):94106). 2000 Mathematics Subject Classification: Primary 26D10; 46E35. Keywords:Hardy inequality, Sobolev Inequality

1. Introduction n ∞n Hardy inequality inℝreads, for all f∈C( R  2  2 (n−2)f 2 |∇f|dx≥dx 2 4|x| n n

andn≥3,

∞n The Sobolev inequality states that, for all f∈C( R 2     ∗ 2 ∗ 2 2 |∇f|dx≥S n |f|dx n n R R

andn≥3,

(1:1)

(1:2)

2n ∗ where 2 = the best constant (cf. [1,2]). Aand is n n −2 S n =πn(n−2)(( )/(n)) 2 (n−2) result of Stubbe [3] states that for , 0≤δ <

n−1   2 (n−2) n 2   −δ    ∗ 2 4 2 f ∗ 2 2   |∇f|dx−δdx≥S n |f|dx n−1 2 |x| n n n R R 2 R (n−2) n 4

(1:3)

and the constant in (1.3) is sharp. Recently, Balinsky et al. [4] prove analogous 2 inequalities for the operator :=x∙ . One of the results states that, for 0≤δ<n/4 ∞n and f∈C( R , 2 n−1        ∗ 2 2 n ∗ 2 2 n 2 (1:4) | L f|dx−δfdx≥C−δS n |rF|dx 4 n n n R R R

© 2011 Xiao; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.