Articles

BURKHOLDER-GUNDY-DAVIS INEQUALITY IN MARTINGALE HARDY SPACES WITH VARIABLE EXPONENT

  • Peide LIU ,
  • Maofa WANG
Expand
  • School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

Received date: 2017-06-30

  Revised date: 2017-12-07

  Online published: 2018-08-25

Supported by

The first author was supported by NSFC (11471251). The second author was supported by NSFC (11271293).

Abstract

In this article, by extending classical Dellacherie's theorem on stochastic sequences to variable exponent spaces, we prove that the famous Burkholder-Gundy-Davis inequality holds for martingales in variable exponent Hardy spaces. We also obtain the variable exponent analogues of several martingale inequalities in classical theory, including convexity lemma, Chevalier's inequality and the equivalence of two kinds of martingale spaces with predictable control. Moreover, under the regular condition on σ-algebra sequence we prove the equivalence between five kinds of variable exponent martingale Hardy spaces.

Cite this article

Peide LIU , Maofa WANG . BURKHOLDER-GUNDY-DAVIS INEQUALITY IN MARTINGALE HARDY SPACES WITH VARIABLE EXPONENT[J]. Acta mathematica scientia, Series B, 2018 , 38(4) : 1151 -1162 . DOI: 10.1016/S0252-9602(18)30805-1

References

[1] Aoyama H. Lebesgue spaces with variable exponent on a probability space. Hiroshima Math J, 2009, 39:207-216
[2] Burkholder D L. Distribution function inequalities for martingales. Ann Prob, 1973, 1:19-42
[3] Burkholder D L, Davis B J, Gundy R F. Integral inequalities for convex functions of operators on martingales. Proc Sixth Berkeley Symp Math Stat and Prob, 1972:223-240
[4] Chao J A, Long R L. Martingale transforms with unbounded multipliers. Proc Amer Math Soc, 1992, 114:831-838
[5] Chevalier L. Un nouveau type d'inegalites pour les martingales discretes. Z Wahrs Verw Gebiete, 1979, 49:249-255
[6] Cruz-Uribe D, Fiorenza A, Martell J M, Perez C. The boundedness of classical operators on variable Lp spaces. Ann Acad Sci Fenn Math, 2006, 31:239-264
[7] Cruz-Uribe D, Fiorenza A, Neugebauer C J. The maximal function on variable Lp(·) spaces. Ann Acad Sci Fenn Math, 2003, 28:223-238; 2004, 29:247-249
[8] Davis B J. On the integrability of the martingale square function. Israel J Math, 1970, 8:187-190
[9] Dellacherie C. Inegalites de convexite pour les processus croissants et les sousmartingales//Sem Prob, LNM 721. Berlin:Springer-Verlag, 1979, 13:371-377
[10] L. Diening, Maximal function on generalized Lebesgue spaces Lp(·). Math Inequal Appl, 2004, 7:245-253
[11] Fan X, Zhao D. On the spaces Lp(x) and Wm,p(x). J Math Anal Appl, 2011, 263:424-446
[12] Garsia A M. Martingale Inequalities:Seminar Notes on Recent Progress. San Diego:Univ California, 1973
[13] Harjulehto P, Hasto P, Pere M. Variable exponent Lebesgue spaces on metric spaces:the Hardy-Littlewood maximal operator. Real Anal Exchange, 2004, 30:87-104
[14] Hudzik H, Kowalewski W. On some global and local geometry properties of Musielak-Orlicz spaces. Publ Math Debrecen, 2005, 67:41-64
[15] Jiao Y, Zhou D J, Hao Z W, Chen W. Martingale Hardy spaces with variable exponent. Banach J Math Anal, 2016, 10:750-770
[16] Kovacik O, Rakosnik J. On spaces Lp(x) and Wk,p(x). Czechoslovak Math J, 1991, 41:592-618
[17] Lenglart E, Lepingle D, Pratelli M. Presentation unifiee de certaines inegalites de la theorie des martigales//Sem Prob, LNM 781. Berlin:Springer-Verlag, 1980, 14:26-48
[18] Lerner A K. Some remarks on the Hardy-Littlewood maximal function on variable Lp spaces. Math Z, 2005, 251:509-521
[19] Long R L. Martingale Spaces and Inequalities. Beijing:Peking Univ Press, 1993
[20] Musielak J. Orlicz Spaces and Modular Spaces. LNM 1034. Berlin:Springer-Verlag, 1983
[21] Nakai E, Sadasue G. Maximal function on generalized martingale Lebesgue spaces with variable exponent. Statis Prob Letters, 2013, 83:2167-2171
[22] Nakai E, Sadasue G. Martingale Morrey-Campanato spaces and fractional integrals. J Funct Spaces Appl, 2012, 673929
[23] Weisz F. Martingale Hardy Spaces and Their Applications in Fourier Analysis. LNM 1568. Berlin:SpringerVerlag, 1994
Options
Outlines

/