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Weighted inequalities for the two-dimensional one-sided Hardy-Littlewood maximal function


Authors: L. Forzani, F. J. Martín-Reyes and S. Ombrosi
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
Published electronically: November 1, 2010
MathSciNet review: 2746661
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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.


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  • 1. K. Andersen and B. Muckenhoupt, Weighted weak type Hardy inequalities with applications to Hilbert transforms and maximal functions, Studia Math. 72 (1982), no. 1, 9-26. MR 665888 (83k:42018)
  • 2. E. 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 (98b:42011)
  • 3. A.L. Bernardis, M. Lorente, F.J. Martín-Reyes, M.T. Martínez and A. de la Torre, Differences of ergodic averages for Cesàro bounded operators, Q. J. Math. 58 (2007), no. 2, 137-150. MR 2334858 (2009g:47027)
  • 4. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241-250. MR 0358205 (50:10670)
  • 5. R. Coifman, P.W. Jones and J.L. Rubio de Francia, Constructive decomposition of BMO functions and factorization of $ A\sb{p}$ weights, Proc. Amer. Math. Soc. 87 (1983), no. 4, 675-676. MR 687639 (84c:42031)
  • 6. N. Dunford and J.T. Schwartz, Linear Operators. I, Pure and Applied Mathematics, Vol. 7 Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London 1958. MR 0117523 (22:8302)
  • 7. J. García-Cuerva and J.L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, 116. Notas de Matemática [Mathematical Notes], 104. North-Holland Publishing Co., Amsterdam, 1985. MR 807149 (87d:42023)
  • 8. T.A. Gillespie and J.L. Torrea, Weighted ergodic theory and dimension free estimates., Q. J. Math. 54 (2003), no. 3, 257-280. MR 2013139 (2004k:47015)
  • 9. M. Lorente and F.J. Martín-Reyes, The ergodic Hilbert transform for Cesàro bounded flows, Tohoku Math. J. (2) 46 (1994), no. 4, 541-556. MR 1301288 (95k:28037)
  • 10. 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), 237-250, Stud. Adv. Math., CRC, Boca Raton, FL, 1995. MR 1330244 (96f:42023)
  • 11. 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 (89i:47015)
  • 12. B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207-226. MR 0293384 (45:2461)
  • 13. S. Ombrosi, Weak weighted inequalities for a dyadic one-sided maximal function in $ \mathbb{R}\sp n$, Proc. Amer. Math. Soc. 133 (2005), no. 6, 1769-1775. MR 2120277 (2005k:42055)
  • 14. R. Sato, A remark on the ergodic Hilbert transform, Math. J. Okayama Univ. 28 (1986), 159-163 (1987). MR 885025 (88k:47043)
  • 15. R. Sato, An ergodic maximal equality for nonsingular flows and applications, Acta Math. Hungar. 61 (1993), no. 1-2, 51-58. MR 1200957 (94c:28020)
  • 16. E. Sawyer, Weighted inequalities for the two-dimensional Hardy operator, Studia Math. 82 (1985), no. 1, 1-16. MR 809769 (87f:42052)
  • 17. E. Sawyer, Weighted inequalities for the one-sided Hardy-Littlewood maximal functions, Trans. Amer. Math. Soc. 297 (1986), no. 1, 53-61. MR 849466 (87k:42018)
  • 18. E. Stein and R. Shakarchi, Real analysis. Measure theory, integration, and Hilbert spaces, Princeton Lectures in Analysis, III. Princeton University Press, Princeton, NJ, 2005. MR 2129625 (2005k:28024)

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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
Email: sombrosi@uns.edu.ar

DOI: https://doi.org/10.1090/S0002-9947-2010-05343-7
Keywords: Weights, one-sided maximal function, ergodic maximal theorem
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
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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