## 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

## 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 â€śtesting type,â€ť 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 â€ścubesâ€ť. We also study some pointwise conditions which are sufficient for strong-type estimates and have applications to solvability of certain nonlinear equations.## References

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## Additional Information

**Igor E. Verbitsky**- Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
- Email: igor@math.missouri.edu
**Richard L. Wheeden**- Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903
- Email: wheeden@math.rutgers.edu
- 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: https://doi.org/10.1090/S0002-9947-98-02017-0
- MathSciNet review: 1443202