Integral Operators on the Cone of Monotone Functions

Abstract

Necessary and sufficient conditions for the boundedness of linear integral operators from L U P (R+) to L Q W (R+) restricted to the cones of monotone functions are given. In addition a general approach to a number of classical operators is explicitly described. In particular, we determine when the Hardy-Littlewood maximal operator is bounded in the classical Lorentz space Γp(v) consisting of those measurable functions on Rn such that ( ∫ 0 ∞ f ∗ ∗ ( t ) p υ ( t ) d t ) 1 / p < ∞ .