Abstract
We prove a weighted mixed-norm inequality for the Doob maximal operator on a filtered measure space. We also give some characterizations of martingale BMO spaces in the setting of Banach function spaces. The main method is based on the technique of extrapolation on martingale Banach spaces.
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K. Andersen and R. John, Weighted inequalities for vector-valued maximal functions and singular integrals, Studia Math., 69 (1980/81), 19–31
Antipa, A.: Doob's inequality for rearrangement-invariant function spaces. Rev. Roumaine Math. Pures Appl. 35, 101–108 (1990)
Aoyama, H.: Lebesgue spaces with variable exponent on a probability space. Hiroshima Math. J. 39, 207–216 (2009)
C. Bennett and R. Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Inc. (Boston, MA, 1988)
Chen, W., Liu, P.: Weighted inequalities for the generalized maximal operator in martingale spaces. Chin. Ann. Math. Ser. B 32, 781–792 (2011)
Chen, W., Liu, P.: Weighted norm inequalities for multisublinear maximal operator on martingale spaces. Tohoku Math. J. 66, 539–553 ((2), 2014,)
Cruz-Uribe, D., Fiorenza, A.: Variable Lebesgue Spaces. Applied and Numerical Harmonic Analysis, Birkhäuser/Springer (Heidelberg (2013)
Cruz-Uribe, D., Fiorenza, A., Martell, J., Pérez, C.: The boundedness of classical operators on variable \(L^p\) spaces. Ann. Acad. Sci. Fenn. Math. 31, 239–264 (2006)
D. Cruz-Uribe, J. Martell, and C. Pérez, Weights, extrapolation and the theory of Rubio de Francia, Operator Theory: Advances and Applications, vol. 215, Birkhäuser/Springer Basel AG (Basel, 2011)
Cruz-Uribe, D., Wang, L.: Variable Hardy spaces. Indiana Univ. Math. J. 63, 447–493 (2014)
Cruz-Uribe, D., Wang, L.: Extrapolation and weighted norm inequalities in the variable Lebesgue spaces. Trans. Amer. Math. Soc. 369, 1205–1235 (2017)
Diening, L.: Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces. Bull. Sci. Math. 129, 657–700 (2005)
L. Diening, R. Harjulehto, P. Hästö, and M. Rǔzička, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics, vol. 2017, Springer (Heidelberg, 2011)
Fefferman, C., Stein, E.M.: Some maximal inequalities. Amer. J. Math. 93, 107–115 (1971)
J. García-Cuerva and J. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, vol. 116, North-Holland Publishing Co. (Amsterdam, 1985)
L. Grafakos, Modern Fourier Analysis, Second ed., Graduate Texts in Mathematics, 250, Springer (New York, 2009)
Ho, K.-P.: Atomic decompositions, dual spaces and interpolations of martingale Hardy-Lorenta-Karamata spaces. Q. J. Math. 65, 985–1009 (2014)
Ho, K.-P.: Atomic decomposition of Hardy spaces and characterization of BMO via Banach function spaces. Anal. Math. f38, 173–185 (2012)
Ho, K.-P.: John-Nirenberg inequalities on Lebesgue spaces with variable exponents. Taiwanese J. Math. 18, 1107–1118 (2014)
Ho, K.-P.: Vector-valued John-Nirenberg inequalities and vector-valued mean osciallations characterization of BMO. Results. Math. 70, 257–270 (2016)
Ho, K.-P.: Strong maximal operator on mixed-norm spaces. Ann. Univ. Ferrara Sez. VII Sci. Mat. 62, 275–291 (2016)
Hytönen, T., Kemppainen, M.: On the relation of Carleson's embedding and the maximal theorem in the context of Banach space geometry. Math. Scand. 109, 269–284 (2011)
T. Hytönen, J. Neerven, M. Veraar, and L. Weis, Analysis in Banach Spaces, Vol. I, Martingales and Littlewood–Paley theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, vol. 63, Springer (Cham, 2016.)
Izuki, M., Nakai, E., Sawano, Y.: Wavelet characterization and modular inequalities for weighted Lebesgue spaces with variable exponent. Ann. Acad. Sci. Fenn. Math. 40, 551–571 (2015)
Izumisawa, M., Kazamaki, N.: Weighted norm inequalities for martingales. Tôhoku Math. J. 29, 115–124 (1977)
Jawerth, B.: Weighted inequalities for maximal operators: linearization, localization and factorization. Amer. J. Math. 108, 361–414 (1986)
Jiao, Y., Peng, L., Liu, P.: Atomic decompositions of Lorentz martingale spaces and applications. J. Funct. Space Appl. 7, 153–166 (2009)
Jiao, Y., Wu, L., Popa, M.: Operator-valued martingale transforms in rearrangement invariant spaces and applications. Sci. China Math. 56, 831–844 (2013)
Jiao, Y., Wu, L., Yang, A., Yi, R.: The predual and John-Nirenberg inequalities on generalized BMO martingale spaces. Trans. Amer. Math. Soc. 369, 537–553 (2017)
Jiao, J., Xie, G., Zhou, D.: Dual spaces and John-Nirenberg inequalities on martingale Hardy-Lorenta-Karamata spaces. Q. J. Math. 66, 605–623 (2015)
Jiao, J., Zhou, D., Hao, Z., Chen, W.: Martingale Hardy spaces with variable exponents. Banach J. Math. Anal. 10, 750–770 (2016)
Kikuchi, M.: Characterization of Banach function space that preserve the Burkholder-square-function inequality. Illinois J. Math. 47, 867–882 (2003)
Kikuchi, M.: On the Davis inequality in Banach function space. Math. Nachr. 281, 697–708 (2008)
Kikuchi, M.: On some inequalities for Doob decompositions in Banach function spaces. Math. Z. 265, 865–887 (2010)
Kurtz, D.: Classical operators on mixed-normed spaces with product weights. Rocky Mountain J. Math. 37, 269–283 (2007)
A. Lerner, On a dual property of the maximal operator on weighted variable \(L^ p\) spaces (preprint), arXiv:1509.07664 (2015)
J. Lindenstrauss and L. Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 97, Springer-Verlag (Berlin–New York, 1979)
R. Long, Martingale Spaces and Inequalities, Peking University Press (Beijing); Friedr. Vieweg & Sohn (Braunschweig, 1993)
Miyamoto, T., Nakai, E., Sadasue, G.: Martingale Orlicz-Hardy spaces. Math. Nachr. 285, 670–686 (2012)
Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc. 165, 207–226 (1972)
E. Nakai and G. Sadasue, Martingale Morrey–Campanato spaces and fractional integrals, J. Funct. Spaces Appl. (2012), Article ID 673929, 29 pp
Nakai, E., Sadasue, G.: Maximal function on generalized martingale Lebesgue space with variable exponent. Stat. Probab. Lett. 83, 2168–2171 (2013)
E. Nakai, G. Sadasue, and Y. Sawano, Martingale Morrey–Hardy and Campanato–Hardy spaces, J. Funct. Spaces Appl., (2013), Article ID 690258, 14 pp
Nakai, E., Sawano, Y.: Hardy spaces with variable exponents and generalized Campanato spaces. J. Funct. Anal. 262, 3665–3748 (2012)
Sawyer, E.: A characterization of a two-weight norm inequality for maximal operators. Studia Math. 75, 1–11 (1982)
Tanaka, H., Terasawa, Y.: Positive operators and maximal operators in a filtered measure space. J. Funct. Anal. 264, 920–946 (2013)
Weisz, F.: Dual spaces of multi-parameter martingale Hardy spaces. Q. J. Math. 67, 137–145 (2016)
F. Weisz, Martingale Hardy Spaces and their Applications in Fourier Analysis, Lecture Notes in Mathematics, vol. 1568, Springer-Verlag (Berlin, 1994)
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The authors thank the referee for careful reading and useful comments.
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Wei Chen is supported by the National Natural Science Foundation of China (11771379), the Natural Science Foundation of Jiangsu Province (BK20161326), the Jiangsu Government Scholarship for Overseas Studies (JS-2017-228) and the School Foundation of Yangzhou University (2016CXJ001).
Dejian Zhou is supported by the National Natural Science Foundation of China (11801573).
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Chen, W., Ho, KP., Jiao, Y. et al. Weighted mixed-norm inequality on Doob’s maximal operator and John–Nirenberg inequalities in Banach function spaces. Acta Math. Hungar. 157, 408–433 (2019). https://doi.org/10.1007/s10474-018-0889-5
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DOI: https://doi.org/10.1007/s10474-018-0889-5