On exact analogues of the Hardy inequality for differences in the case of related weights

The paper deals with Hardy-type inequalities when the right-hand side contains the weighted norm of difference, modulus of continuity or the best approximation operator of a given function. The authors give necessary and sufficient conditions for the case of related weights. The following result is typical. par Theorem 1. Let 1le p <infty. Suppose a given weight wge 0 is decreasing on [0,infty) and v(x)=int_x^infty w<infty for all x>0. Then the inequality multline int_0^a |f|^p v le Cleft( v(a)int_0^a |f|^p + int_0^aint_0^a |f(x) - f(y)|^p w(|x-y|)dxdyright) endmultline holds if and only if int_0^t v le Ctv(t).