Exact analogues of the Hardy inequality for differences in the case of related weights. (Q1594390)

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. Theorem 1. Let \(1\leq p <\infty\). Suppose a given weight \(w\geq 0\) is decreasing on \([0,\infty)\) and \(v(x)=\int_x^\infty w< \infty\) for all \(x>0\). Then the inequality \[ \int_0^a| f|^p v\leq C\left( v(a)\int_0^a | f|^p + \int_0^a\int_0^a | f(x) - f(y)|^p w(| x-y|)\,dxdy\right) \] holds if and only if \(\int_0^t v \leq Ctv (t)\).