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

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.