Weighted norm inequalities for integral operators |
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Authors: | Igor E Verbitsky Richard L Wheeden |
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Institution: | Department of Mathematics, University of Missouri, Columbia, Missouri 65211 ; Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903 |
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Abstract: | We consider a large class of positive integral operators acting on functions which are defined on a space of homogeneous type with a group structure. We show that any such operator has a discrete (dyadic) version which is always essentially equivalent in norm to the original operator. As an application, we study conditions of ``testing type,' like those initially introduced by E. Sawyer in relation to the Hardy-Littlewood maximal function, which determine when a positive integral operator satisfies two-weight weak-type or strong-type estimates. We show that in such a space it is possible to characterize these estimates by testing them only over ``cubes'. We also study some pointwise conditions which are sufficient for strong-type estimates and have applications to solvability of certain nonlinear equations. |
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