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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Weighted norm inequalities for integral operators
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by Igor E. Verbitsky and Richard L. Wheeden PDF
Trans. Amer. Math. Soc. 350 (1998), 3371-3391 Request permission


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
  • Email:
  • Richard L. Wheeden
  • Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903
  • Email:
  • 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.
  • © Copyright 1998 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 350 (1998), 3371-3391
  • MSC (1991): Primary 42B20, 42B25
  • DOI:
  • MathSciNet review: 1443202