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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Weighted norm inequalities for integral operators

Authors: Igor E. Verbitsky and Richard L. Wheeden
Journal: Trans. Amer. Math. Soc. 350 (1998), 3371-3391
MSC (1991): Primary 42B20, 42B25
MathSciNet review: 1443202
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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 $(L^{p}, L^{q})$ 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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Additional Information

Igor E. Verbitsky
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211

Richard L. Wheeden
Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903

Received by editor(s): March 28, 1996
Received by editor(s) in revised form: October 1, 1996
Additional Notes: The first author was partially supported by NSF Grant DMS94-01493 and the second by NSF Grant DMS95-00799.
Article copyright: © Copyright 1998 American Mathematical Society