Generalized Hardy operators and normalizing measures

Necessary and sufficient conditions on the weight nu and the measure sigma for the operator Kf(s) = integral(a(s))(b(s))k(s,y)f(y)dy to be bounded from L-nu(p)[0,infinity) to L-sigma(q)(S) are given. Here a(s) and b(s) are similarly ordered functions and k(s,y) satisfies a modified GHO condition. Nearly block diagonal decompositions of positive operators are introduced as is the concept of a normalizing measure. An application is made to estimates for the remainder in a Taylor approximation.