Journal of Inequalities and Applications
Volume 1 (1997), Issue 1, Pages 1-10
doi:10.1155/S1025583497000015

On weighted hardy and Poincaré-type inequalities for differences

V. I. Burenkov,1 W. D. Evans,1 and M. L. Goldman2

1School of Mathematics, University of Wales, Cardiff, Senghennydd Road, Cardiff CF2 4AG, UK
2Moscow State Inst. of Radio Engineering, Electronics and Automation (Tech. Univ.), Pr. Vernadskogo 78, Moscow, Russia

Received 7 June 1996

Copyright © 1997 V. I. Burenkov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A criterion is obtained for the Hardy-type inequality (∫0a|f(x)|pv(x)dx)1/p≤c1{(v(a)∫0a|f(x)|pdx)1/p+(∫0a∫0a|f(x)−f(y)|pw(|x−y|)dxdy)1/p} to be valid for 0<a≤A≤∞ and 0<p<∞. This weakens a criterion previously found by the first two authors and comes close to being necessary as well as sufficient. When an inequality in the criterion is reversed, a Poincaré-type inequality is derived in some cases.