Journal of Inequalities and Applications 
Volume 1 (1997), Issue 2, Pages 183-197
doi:10.1155/S1025583497000131

Weighted modular inequalities for monotone functions

P. Drábek,1 H. P. Heinig,2 and A. Kufner3

 1Department of Mathematics, University of West Bohemia, P.O. Box 314, Pilsen 306 14, Czech Republic
2Department of Mathematics and Statistics, McMaster University, Ontario, Hamilton L8S 4K1, Canada
3Mathematical Institute, Czech Academy of Sciences, Zitna 25, Prague 11567, Czech Republic
Received 4 June 1996

Abstract

Weight characterizations of weighted modular inequalities for operators on the cone of monotone functions are given in terms of composition operators on arbitrary non-negative functions with changes in weights. The results extend to modular inequalities, those corresponding to weighted Lebesgue spaces given by E.T. Sawyer [15]. Application to Hardy and fractional integral operators on monotone functions are given.