WEIGHTED NORM INEQUALITIES FOR INTEGRAL-OPERATORS

We consider a large class of positive integral operators acting on fun ctions which are defined on a space of homogeneous type with a group s tructure. We show that any such operator has a discrete (dyadic) versi on which is always essentially equivalent in norm to the original oper ator. As an application, we study conditions of ''testing type,'' like those initially introduced by E. Sawyer in relation to the Hardy-Litt lewood maximal function, which determine when a positive integral oper ator satisfies two-weight weak-type or strong-type (L-p, L-q) estimate s. We show that in such a space it is possible to characterize these e stimates by testing them only over ''cubes''. We also study some point wise conditions which are sufficient for strong-type estimates and hav e applications to solvability of certain nonlinear equations.