Criteria of two-weight inequalities for integral transformations with a positive kernel and maximal functions (Q1386588)

In this survey paper the authors characterize boundedness of certain integral operators \(T\) (\(T\) is either an integral transform with a positive kernel, or the \(n\)-dimensional fractional maximal operator, or the fractional integral of Riemann-Liouville or of Weyl type with \(0<\alpha<1\), or the one-dimensional one-sided fractional maximal operator) from a weighted Lebesgue space \(L^p(X,wd\mu)\) into a weighted Lebesgue space \(L^q(X,v d\mu)\) (or a weak weighted Lebesgue space \(L_*^q(X,v d\mu)\)) where \(1<p\leq q<\infty\), \(X=(X,d,\mu)\) is a space of homogeneous type or \(X=(0,\infty)\) with the Lebesgue measure. Authors' results extend the previous ones removing some assumptions on the space \((X,d,\mu)\). In the case when \(T\) is the Riemann-Liouville or the Weyl fractional integral, characterization was previously known only for \(\alpha\geq 1\). All the results are presented without proofs.