On weighted hardy and Poincaré-type inequalities for differences

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