DOOB'S INEQUALITY, BURKHOLDER-GUNDY INEQUALITY AND MARTINGALE TRANSFORMS ON MARTINGALE MORREY SPACES

doi:10.1016/S0252-9602(17)30119-4

摘要/Abstract

摘要：

We introduce the martingale Morrey spaces built on Banach function spaces. We establish the Doob's inequality, the Burkholder-Gundy inequality and the boundedness of martingale transforms for our martingale Morrey spaces. We also introduce the martingale block spaces. By the Doob's inequality on martingale block spaces, we obtain the Davis' decompositions for martingale Morrey spaces.

关键词: Morrey spaces,Banach function space, block spaces,Doob's inequality, Burkholder-Gundy inequality, martingale transform, Davis'decomposition, martingale

Abstract:

Key words: Morrey spaces,Banach function space, block spaces,Doob's inequality, Burkholder-Gundy inequality, martingale transform, Davis'decomposition, martingale

Kwok-Pun HO. DOOB'S INEQUALITY, BURKHOLDER-GUNDY INEQUALITY AND MARTINGALE TRANSFORMS ON MARTINGALE MORREY SPACES[J]. 数学物理学报(英文版), 2018, 38(1): 93-109.

Kwok-Pun HO. DOOB'S INEQUALITY, BURKHOLDER-GUNDY INEQUALITY AND MARTINGALE TRANSFORMS ON MARTINGALE MORREY SPACES[J]. Acta mathematica scientia,Series B, 2018, 38(1): 93-109.

参考文献

[1] Antipa A. Doob's inequality for rearrangement-invariant function spaces. Rev Roumaine Math Pures Appl, 1990, 35:101-108
[2] Aoyama H. Lebesgue spaces with variable exponent on a probability space. Hiroshima Math J, 2009, 39:207-216
[3] Bennett C, Sharpley R. Interpolations of Operators. New York:Academic Press, 1988
[4] Blasco O, Ruiz A, Vega L. Non interpolation in Morrey-Campanato and Block Spaces. Ann Scuola Norm Sup Pisa Cl Sci, 1999, 28(4):31-40
[5] Burkholder D. Martingale transforms. Ann Math Stat, 1966, 37:1495-1504
[6] Doob J. Semimartingales and subharmonic functions. Trans Amer Math Soc, 1954, 77:86-121
[7] Garsia A. Martingale Inequalities, Seminar Notes on Recent Progress. Math Lecture Note Series, New York, Benjamin Inc, 1973
[8] Herz C. Bounded mean oscillation and regulated martingales. Trans Amer Math Soc, 1974, 193:199-215
[9] Ho K-P. Littlewood-Paley Spaces. Math Scand, 2011, 108:77-102
[10] Ho K-P. Vector-valued singular integral operators on Morrey type spaces and variable Triebel-LizorkinMorrey spaces. Ann Acad Sci Fenn Math, 2012, 37:375-406
[11] Ho K-P. Atomic decomposition of Hardy spaces and characterization of BMO via Banach function spaces. Anal Math, 2012, 38:173-185
[12] Ho K-P. The fractional integral operators on Morrey spaces with variable exponent on unbounded domains. Math Inequal Appl, 2013, 16:363-373
[13] Ho K-P. Atomic decompositions, dual spaces and interpolations of martingale Hardy-Lorenta-Karamata spaces. Q J Math, 2014, 65:985-1009
[14] Ho K-P. Vector-valued operators with singular kernel and Triebel-Lizorkin-block spaces with variable exponents. Kyoto J Math, 2016, 56:97-124
[15] Ho K-P. Atomic decompositions of martingale Hardy-Morrey spaces. Acta Math Hungar, 2016, 149:177-189
[16] Ho K-P. Two-weight norm, Poincaré, Sobolev and Stein Weiss inequalities on Morrey spaces. Publ Res Inst Math Sci, 2017, 53:119-139
[17] Ho K-P. Fractional integral operators with homogeneous kernels on Morrey spaces with variable exponents. J Math Soc Jpn, 2017, 69:1059-1077
[18] Ho K-P. Extrapolation, John-Nirenberg inequalities and characterizations of BMO in terms of Morrey type spaces. Rev Mat Complut, 2017, 30:487-505
[19] Jiao Y, Peng L, Liu P. Atomic decompositions of Lorentz martingale spaces and applications. J Funct Space Appl, 2009, 7:153-166
[20] Jiao Y, Wu L, Popa M. Operator-valued martingale transforms in rearrangement-invariant spaces and applications. Sci China Math, 2013, 56:831-844
[21] Jiao Y, Xie G, Zhou D. Dual spaces and John-Nirenberg inequalities on martingale Hardy-LorentaKaramata spaces. Q J Math, 2015, 66:605-623
[22] Kikuchi M. Characterization of Banach function space that preserve the Burkholder-square-function inequality. Illinois J Math, 2003, 47:867-882
[23] Kikuchi M. On the Davis inequality in Banach function space. Math Nachr, 2008, 281:697-708
[24] Kikuchi M. On some inequalities for Doob decompositions in Banach function spaces. Math Z, 2010, 265:865-887
[25] Long R. Martingale Spaces and Inequalities. Beijing:Peking University Press, 1993
[26] Meyer Y, Taibleson M, Weiss G. Some functional analytic properties of the Space B_q generated by blocks. Indiana Univ Math J, 1985, 34:493-515
[27] Miyamoto T, Nakai E, Sadasue G. Martingale Orlicz-Hardy spaces. Math Nachr, 2012, 285:670-686
[28] Morrey C. On the solutions of quasi-linear elliptic partial differential equations. Trans Amer Math Soc, 1938, 43:126-166
[29] Nakai E. Hardy-Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces. Math Nachr, 1994, 166:95-104
[30] Nakai E. Orlicz-Morrey spaces and the Hardy-Littlewood maximal function. Studia Math, 2008, 188:193-221
[31] Nakai E, Sadasue G. Martingale Morrey-Campanato spaces and fractional integrals. J Funct Spaces Appl, 2012, Article ID:673929, 29 pages
[32] Nakai E, Sadasue G. Maximal function on generalized martingale Lebesgue space with variable exponent. Stat Probab Lett, 2013, 83:2168-2171
[33] Nakai E, Sadasue G, Sawano Y. Martingale Morrey-Hardy and Campanato-Hardy spaces. J Funct Spaces Appl, 2013, Article ID:690258, 14 pages
[34] Soria F. Characterizations of classes of functions generated by blocks and associated Hardy spaces. Indiana Univ Math J, 1985, 34:463-492
[35] Stein E. Topics in Harmonic Analysis Related to the Littlewood-Paley Theory. Annals of Mathematics Studies Number 63, Princeton University Press, 1970
[36] Weisz F. Martingale Hardy spaces for 0 < p ≤ 1. Probab Th Rel Fields, 1990, 84:361-376
[37] Weisz F. Martingale Hardy Spaces and Their Applications in Fourier Analysis. Lecture Notes in Maths Vol 1568, Springer-Verlag, 1994
[38] Weisz F. Bounded linear operator on weak Hardy spaces and applications. Acta Math Hung, 1998, 80:249-264