Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-98-02017-0
MathSciNet review: 1443202
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 42B20, 42B25

Retrieve articles in all journals with MSC (1991): 42B20, 42B25


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.
Article copyright: © Copyright 1998 American Mathematical Society