On weighted hardy and Poincaré-type inequalities for differences

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.

Keywords

• Hardy; Poincar&#233;; inequalities; differences.