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 $(L^{p}, L^{q})$ 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.
- Michael Christ, A $T(b)$ theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (1990), no. 2, 601–628. MR 1096400, DOI https://doi.org/10.4064/cm-60-61-2-601-628
- Ronald R. Coifman and Guido Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), no. 4, 569–645. MR 447954, DOI https://doi.org/10.1090/S0002-9904-1977-14325-5
- N. J. Kalton and I. E. Verbitsky, Nonlinear equations and weighted norm inequalities, Trans. Amer. Math. Soc. (to appear).
- Vladimir G. Maz′ya and Igor E. Verbitsky, Capacitary inequalities for fractional integrals, with applications to partial differential equations and Sobolev multipliers, Ark. Mat. 33 (1995), no. 1, 81–115. MR 1340271, DOI https://doi.org/10.1007/BF02559606
- Eric T. Sawyer, A characterization of a two-weight norm inequality for maximal operators, Studia Math. 75 (1982), no. 1, 1–11. MR 676801, DOI https://doi.org/10.4064/sm-75-1-1-11
- Eric Sawyer, A two weight weak type inequality for fractional integrals, Trans. Amer. Math. Soc. 281 (1984), no. 1, 339–345. MR 719674, DOI https://doi.org/10.1090/S0002-9947-1984-0719674-6
- Eric T. Sawyer, A characterization of two weight norm inequalities for fractional and Poisson integrals, Trans. Amer. Math. Soc. 308 (1988), no. 2, 533–545. MR 930072, DOI https://doi.org/10.1090/S0002-9947-1988-0930072-6
- E. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math. 114 (1992), no. 4, 813–874. MR 1175693, DOI https://doi.org/10.2307/2374799
- E. T. Sawyer, R. L. Wheeden, and S. Zhao, Weighted norm inequalities for operators of potential type and fractional maximal functions, Potential Analysis 5 (1996), 523–580.
- I. E. Verbitsky and R. L. Wheeden, Weighted trace inequalities for fractional integrals and applications to semilinear equations, J. Funct. Anal. 129 (1995), no. 1, 221–241. MR 1322649, DOI https://doi.org/10.1006/jfan.1995.1049
- Richard L. Wheeden and Shiying Zhao, Weak type estimates for operators of potential type, Studia Math. 119 (1996), no. 2, 149–160. MR 1391473
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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.
Article copyright:
© Copyright 1998
American Mathematical Society
