Hardy Inequalities on Cones of Monotone Functions on a Measure Space

Consider a Hausdorff topological space X with Borel measure .Let { (t)} be a totally ordered family of open subsets of an open set in X parameterized by t [0, ).This family generates a partial order in the set of functions defined on .We obtain Sawyer-type bounds for a linear functional on cones of decreasing and increasing functions in weighted spaces.These results are further applied to obtain criteria of boundedness of averaging operators in those weighted spaces.