$A_p$ weights for nondoubling measures in $R^n$ and applications
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- by Joan Orobitg and Carlos Pérez PDF
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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 $\frac {1}{\mu (Q)}\int _{Q} |f-f_{Q}| d\mu \le a(Q)$ 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 $a$.References
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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
- 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. - © Copyright 2002 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 354 (2002), 2013-2033
- MSC (2000): Primary 42B25, 42B20
- DOI: https://doi.org/10.1090/S0002-9947-02-02922-7
- MathSciNet review: 1881028