Reduction theorems for weighted integral inequalities on the cone of monotone functions (Q2868456)

This paper gives an excellent survey of weighted integral inequalities in Lebesgue spaces on cones of monotone functions on the real semi-axis. More specifically, surveys of results related to the reduction of integral inequalities involving positive operators in weighted Lebesgue spaces on the real semi-axis and valid on the cone of monotone functions are carried out, discussed and in some cases new proofs are provided. Furthermore, a new method of reduction of \((L_{p,v}^{\downarrow }\rightarrow L_{q,w}^{+})\)- and \((L_{p,v}^{\uparrow }\rightarrow L_{q,w}^{+})\)-inequalities for positive monotone operators, where \(1\leq p\leq \infty \) and \( 0<q\leq \infty\), are derived and proved. In particular, this method helps to reduce a given inequality on monotone functions to some inequality on nonnegative functions which is easier characterized than the original inequality. The problem for the integral Volterra operator for the case \(0<q<p\leq 1\) and its dual with a kernel satisfying Oinarov's condition are solved and discussed. As application, a complete characterization for all possible integrability parameters are obtained for a number of Volterra operators.