Weighted norm inequalities for averaging operators of monotone functions (Q1182633)

The averaging operator \(A\) is defined by \(Af(x)=x^{-1}\int_ 0^ x f(x)dx\). The author investigates for which weight pair \((w,v)\) the weighted inequality \[ \int_ 0^ \infty| Af|^ p w(x)dx\leq C\int_ 0^ \infty| f|^ p v(x)dx\leqno (*) \] holds. Muckenhoupt characterized \((w,w)\) for which \((*)\) holds for general \(f\), and Ariño and Muckenhoupt treated the case for nonincreasing \(f\). In this case, the \(B_ p\) condition appeared, i.e., \[ \int_ r^ \infty(x/r)^ p w(x)dx\leq C\int_ 0^ r w(x)dx, \qquad r>0. \] The author gives a more general version of \((*)\), and as a special case gives a shorter proof of the Ariño and Muckenhoupt result. The nondecreasing case and other related results are treated.