Weighted inequalities for the two-dimensional one-sided Hardy-Littlewood maximal function
HTML articles powered by AMS MathViewer
- by L. Forzani, F. J. Martín-Reyes and S. Ombrosi PDF
- Trans. Amer. Math. Soc. 363 (2011), 1699-1719 Request permission
Abstract:
In this work we characterize the pairs of weights $(w,v)$ such that the one-sided Hardy-Littlewood maximal function in dimension two is of weak-type $(p,p)$, $1\leq p<\infty$, with respect to the pair $(w,v)$. As an application of this result we obtain a generalization of the classic Dunford-Schwartz Ergodic Maximal Theorem for bi-parameter flows of null-preserving transformations.References
- Kenneth F. Andersen and Benjamin Muckenhoupt, Weighted weak type Hardy inequalities with applications to Hilbert transforms and maximal functions, Studia Math. 72 (1982), no. 1, 9–26. MR 665888, DOI 10.4064/sm-72-1-9-26
- Earl Berkson and T. A. Gillespie, Mean-boundedness and Littlewood-Paley for separation-preserving operators, Trans. Amer. Math. Soc. 349 (1997), no. 3, 1169–1189. MR 1407694, DOI 10.1090/S0002-9947-97-01896-5
- A. L. Bernardis, M. Lorente, F. J. Martín-Reyes, A. De La Torre, and M. T. Martínez, Differences of ergodic averages for Cesàro bounded operators, Q. J. Math. 58 (2007), no. 2, 137–150. MR 2334858, DOI 10.1093/qmath/hal023
- R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250. MR 358205, DOI 10.4064/sm-51-3-241-250
- R. Coifman, Peter W. Jones, and José L. Rubio de Francia, Constructive decomposition of BMO functions and factorization of $A_{p}$ weights, Proc. Amer. Math. Soc. 87 (1983), no. 4, 675–676. MR 687639, DOI 10.1090/S0002-9939-1983-0687639-3
- Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
- José García-Cuerva and José L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, vol. 116, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 104. MR 807149
- T. A. Gillespie and José L. Torrea, Weighted ergodic theory and dimension free estimates, Q. J. Math. 54 (2003), no. 3, 257–280. MR 2013139, DOI 10.1093/qjmath/54.3.257
- María Lorente Domínguez and Francisco Javier Martín-Reyes, The ergodic Hilbert transform for Cesàro bounded flows, Tohoku Math. J. (2) 46 (1994), no. 4, 541–556. MR 1301288, DOI 10.2748/tmj/1178225679
- F. J. Martín-Reyes, On the one-sided Hardy-Littlewood maximal function in the real line and in dimensions greater than one, Fourier analysis and partial differential equations (Miraflores de la Sierra, 1992) Stud. Adv. Math., CRC, Boca Raton, FL, 1995, pp. 237–250. MR 1330244
- F. J. Martín-Reyes and A. De la Torre, The dominated ergodic estimate for mean bounded, invertible, positive operators, Proc. Amer. Math. Soc. 104 (1988), no. 1, 69–75. MR 958045, DOI 10.1090/S0002-9939-1988-0958045-3
- Benjamin Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226. MR 293384, DOI 10.1090/S0002-9947-1972-0293384-6
- Sheldy Ombrosi, Weak weighted inequalities for a dyadic one-sided maximal function in $\Bbb R^n$, Proc. Amer. Math. Soc. 133 (2005), no. 6, 1769–1775. MR 2120277, DOI 10.1090/S0002-9939-05-07830-5
- Ryotaro Sato, A remark on the ergodic Hilbert transform, Math. J. Okayama Univ. 28 (1986), 159–163 (1987). MR 885025
- R. Sato, An ergodic maximal equality for nonsingular flows and applications, Acta Math. Hungar. 61 (1993), no. 1-2, 51–58. MR 1200957, DOI 10.1007/BF01872096
- E. Sawyer, Weighted inequalities for the two-dimensional Hardy operator, Studia Math. 82 (1985), no. 1, 1–16. MR 809769, DOI 10.4064/sm-82-1-1-16
- E. Sawyer, Weighted inequalities for the one-sided Hardy-Littlewood maximal functions, Trans. Amer. Math. Soc. 297 (1986), no. 1, 53–61. MR 849466, DOI 10.1090/S0002-9947-1986-0849466-0
- Elias M. Stein and Rami Shakarchi, Real analysis, Princeton Lectures in Analysis, vol. 3, Princeton University Press, Princeton, NJ, 2005. Measure theory, integration, and Hilbert spaces. MR 2129625
Additional Information
- L. Forzani
- Affiliation: IMAL-CONICET, Facultad de Ingeniería Química, Universidad Nacional del Litoral, Güemes 3450, 3000 Santa Fe, Argentina
- Email: liliana.forzani@gmail.com
- F. J. Martín-Reyes
- Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071-Málaga, Spain
- Email: martin_reyes@uma.es
- S. Ombrosi
- Affiliation: Departamento de Matemática, Universidad Nacional del Sur, Bahía Blanca, 8000, Argentina
- MR Author ID: 713193
- Email: sombrosi@uns.edu.ar
- Received by editor(s): May 28, 2007
- Published electronically: November 1, 2010
- Additional Notes: The research of the first author has been partially supported by CONICET, grant PIP 5810, and the Universidad Nacional del Litoral
The research of the second author has been partially supported by the Junta de Andalucía, grants FQM-354 and the FQM-01509, and the Spanish government, grants MTM2005-08350-C03-02 and MTM2008-06621-C02-02
The research of the third author has been partially supported by the Universidad Nacional del Sur, grant SGCyT-UNS PGI 24/L058, and by a grant of the Spanish government, MEC Res. 26/05/2006 - © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 363 (2011), 1699-1719
- MSC (2010): Primary 42B25; Secondary 47A35, 37A40
- DOI: https://doi.org/10.1090/S0002-9947-2010-05343-7
- MathSciNet review: 2746661