Abstract
We introduce Orlicz–Hardy spaces for two-parameter martingales, and establish some new martingale inequalities by use of the atomic decompositions. We also characterize the dual spaces of martingale Orlicz–Hardy spaces in two-parameter case as the generalized martingale Lipschitz spaces.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
D. Burkholder, R. Gundy, Extrapolation and interpolation of quasi-linear operators on martingales, Acta Math., 124 (1970), 249–304.
R. Cairoli, Une inégalité pour martingale à indices multiples et res applications, in: Séminaire de Probabilités IV, Lecture Notes in Math., vol. 124, Springer (Berlin, 1970), pp.1–27.
R. Cairoli and J. Walsh, Stochastic integrals in the plane, Acta Math., 134 (1975), 111–183.
B. Davis, On the integrability of the martingale square function, Israel J. Math., 8 (1970), 187–190.
J. Doob, Semimartingales and subharmonic functions, Trans. Amer. Math. Soc., 77 (1954), 86–121.
A. Garsia, Martingale Inequalities: Seminar Notes on Recent Progress, Math. Lecture Notes Series, Benjamin Inc. (New York, 1973).
C. Herz, Bounded mean oscillation and regulated martingales, Trans. Amer. Math. Soc., 193 (1974), 199–215.
Y. Jiao, F. Weisz, G. Xie and D. Yang, Martingale Musielak–Orlicz–Lorentz Hardy spaces and applications to dyadic Fourier analysis, J. Geom. Anal. (2021), https://doi.org/10.1007/s12220-021-00671-8.
Y. Jiao, L. Wu, A. Yang and R. Yi, The predual and John–Nirenberg inequalities on generalized BMO martingale spaces, Trans. Amer. Math. Soc., 369 (2017), 537–553.
R. Long, Martingale Spaces and Inequalities, Science Press (Beijing, 1993).
C. Métraux, Quelques inégalités pour martingales à paramétre bidimensionnel, Séminaire de Probabilités XII, Lecture Notes in Math., vol. 649, Springer (Berlin, 1978), pp. 170–179.
T. Miyamoto, E. Nakai and G. Sadasue, Martigale Orlicz–Hardy spaces, Math. Nachr., 285 (2012), 670–686.
M. Mohsenipour and G. Sadeghi, Atomic decomposition of martingale weighted Lorentz spaces with two-parameter and applications, Rocky Mountain J. Math., 47 (2017), 927–945.
E. Nakai and Y. Sawano, Orlicz–Hardy spaces and their duals, Sci. China Math., 57 (2014), 903–962.
J. Peetre, On interpolation functions. II, Acta. Sci. Math. (Szeged), 29 (1968), 91–92.
F. Weisz, Martigale Hardy spaces for \(0 < p \leq \) 1, Probab. Theory Related Fields, 84 (1990), 361–376.
F. Weisz, On duality problems of two-parameter martingale Hardy spaces, Bull. Sci. Math., 114 (1990), 395–410.
F. Weisz, Martigale Hardy spaces and their Applications in Fourier Analysis, Lecture Notes in Math., vol. 1568, Springer-Verlag (Berlin, 1994).
F. Weisz, Dual spaces of multi-parameter martingale Hardy spaces, Quart. J. Math., 67 (2016), 137–145.
G. Xie, Y. Jiao and D. Yang, Martingale Musielak–Orlicz Hardy spaces, Sci. China Math., 62 (2019), 1567–1584.
G. Xie, F. Weisz, D. Yang and Y. Jiao, New martingale inequalities and applications to Fourier analysis, Nonlinear Anal., 182 (2019), 143–192.
Author information
Authors and Affiliations
Corresponding author
Additional information
Jianzhong Lu is supported by the Fundamental Research Funds for the Central University of Central South University (No. 2021zzts0032).
Rights and permissions
About this article
Cite this article
Lu, JZ. Two-parameter martingale Orlicz–Hardy spaces. Acta Math. Hungar. 166, 30–47 (2022). https://doi.org/10.1007/s10474-021-01201-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-021-01201-2