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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