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

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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 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:
igor@math.missouri.edu

**Richard L. Wheeden**

Affiliation:
Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903

Email:
wheeden@math.rutgers.edu

DOI:
https://doi.org/10.1090/S0002-9947-98-02017-0

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