Weighted inequalities for the fractional integral operators on monotone functions (Q1909644)

Summary: We give a characterization of weight functions \(u\) and \(v\) on \(\mathbb{R}^n\) for which the fractional integral operator \(I_s\) of order \(s\) on \(\mathbb{R}^n\) defined by \((I_s f)(x)= \int_{\mathbb{R}^n} |x- y|^{s-n} f(y)dy\) sends all monotone functions which belong to the weighted Lebesgue space \(L^p_v(\mathbb{R}^n)\) into the weighted Lebesgue space \(L^q_u(\mathbb{R}^n)\). This characterization is done for all \(p\) and \(q\) with \(1< p< \infty\) and \(0< q< \infty\). The analogous Lorentz and Orlicz problems are also considered.