Ie. Verbitsky et Rl. Wheeden, WEIGHTED NORM INEQUALITIES FOR INTEGRAL-OPERATORS, Transactions of the American Mathematical Society, 350(8), 1998, pp. 3371-3391
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.