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Reduction theorems for weighted integral inequalities on the cone of monotone functions. (English. Russian summary) Zbl 1288.26018
Russ. Math. Surv. 68, No. 4, 597-664 (2013); translation from Usp. Mat. Nauk 68, No. 4, 3-68 (2013).
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.

MSC:
26D15 Inequalities for sums, series and integrals
47G10 Integral operators
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