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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

$A_p$ weights for nondoubling measures in $R^n$ and applications


Authors: Joan Orobitg and Carlos Pérez
Journal: Trans. Amer. Math. Soc. 354 (2002), 2013-2033
MSC (2000): Primary 42B25, 42B20
Published electronically: January 11, 2002
MathSciNet review: 1881028
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Abstract: We study an analogue of the classical theory of $A_p(\mu)$weights in $\mathbb{R} ^n $ without assuming that the underlying measure $\mu$is doubling. Then, we obtain weighted norm inequalities for the (centered) Hardy-Littlewood maximal function and corresponding weighted estimates for nonclassical Calderón-Zygmund operators. We also consider commutators of those Calderón- Zygmund operators with bounded mean oscillation functions ($BMO$), extending the main result from R. Coifman, R. Rochberg, and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. 103 (1976), 611-635. Finally, we study self-improving properties of Poincaré-B.M.O. type inequalities within this context; more precisely, we show that if $f$ is a locally integrable function satisfying $\frac{1}{\mu(Q)}\int_{Q} \vert f-f_{Q}\vert d\mu \le a(Q)$ for all cubes $Q$, then it is possible to deduce a higher $L^p$ integrability result for $f$, assuming a certain simple geometric condition on the functional $a$.


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Additional Information

Joan Orobitg
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bella- terra, Spain
Email: orobitg@mat.uab.es

Carlos Pérez
Affiliation: Departamento de Matemáticas, Universidad Autónoma de Madird, 28049 Madrid, Spain
Address at time of publication: Department of Mathematical Analysis, Universidad de Sevilla, 41080 Sevilla, Spain
Email: carlos.perez@uam.es

DOI: http://dx.doi.org/10.1090/S0002-9947-02-02922-7
PII: S 0002-9947(02)02922-7
Received by editor(s): February 23, 2000
Received by editor(s) in revised form: September 12, 2000
Published electronically: January 11, 2002
Additional Notes: The first author’s research was partially supported by CIRIT grant 2000 SGR00059 and by DGICYT grant BFM 2000-0361, Spain.
The second author’s research was partially supported by DGESIC grant PB98-0106, Spain.
Article copyright: © Copyright 2002 American Mathematical Society