Weighted \(L^P-L^Q\) inequalities for the fractional maximal operator when \(1<q<p<\infty\) (Q1977167)

For \(0\leq\alpha<n\), the fractional maximal operator \(M_\alpha\) is defined by \[ M_\alpha f(x)=\sup_{t>0}t^{\alpha-n}\int_{|x-y|<t}|f(y)|dy. \] The author gives some necessary conditions and some sufficient conditions on weights \(u(\cdot)\) and \(v(\cdot)\) for which the fractional maximal operator \(M_\alpha\) is bounded from the weighted Lebesgue spaces \(L^p_v({\mathbb R}^n)\) into \(L^p_u({\mathbb R}^n)\) for \(1<q<p<\infty\) and \(0\leq\alpha<n\).