Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-11T01:03:32.978Z Has data issue: false hasContentIssue false
  • English
  • Français

Interpolation between noncommutative martingale Hardy and BMO spaces: the case $\textbf {0<p<1}$

Part of: Stochastic processes Normed linear spaces and Banach spaces; Banach lattices Topological linear spaces and related structures Selfadjoint operator algebras

Published online by Cambridge University Press:  25 August 2021

Narcisse Randrianantoanina
Narcisse Randrianantoanina*
Affiliation:
Department of Mathematics, Miami University, Oxford, OH 45056, USA
*
e-mail: randrin@miamioh.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Let $\mathcal {M}$ be a semifinite von Nemann algebra equipped with an increasing filtration $(\mathcal {M}_n)_{n\geq 1}$ of (semifinite) von Neumann subalgebras of $\mathcal {M}$ . For $0<p <\infty $ , let $\mathsf {h}_p^c(\mathcal {M})$ denote the noncommutative column conditioned martingale Hardy space and $\mathsf {bmo}^c(\mathcal {M})$ denote the column “little” martingale BMO space associated with the filtration $(\mathcal {M}_n)_{n\geq 1}$ .

We prove the following real interpolation identity: if $0<p <\infty $ and $0<\theta <1$ , then for $1/r=(1-\theta )/p$ ,

$$ \begin{align*} \big(\mathsf{h}_p^c(\mathcal{M}), \mathsf{bmo}^c(\mathcal{M})\big)_{\theta, r}=\mathsf{h}_{r}^c(\mathcal{M}), \end{align*} $$
with equivalent quasi norms.

For the case of complex interpolation, we obtain that if $0<p<q<\infty $ and $0<\theta <1$ , then for $1/r =(1-\theta )/p +\theta /q$ ,

$$ \begin{align*} \big[\mathsf{h}_p^c(\mathcal{M}), \mathsf{h}_q^c(\mathcal{M})\big]_{\theta}=\mathsf{h}_{r}^c(\mathcal{M}) \end{align*} $$
with equivalent quasi norms.

These extend previously known results from $p\geq 1$ to the full range $0<p<\infty $ . Other related spaces such as spaces of adapted sequences and Junge’s noncommutative conditioned $L_p$ -spaces are also shown to form interpolation scale for the full range $0<p<\infty $ when either the real method or the complex method is used. Our method of proof is based on a new algebraic atomic decomposition for Orlicz space version of Junge’s noncommutative conditioned $L_p$ -spaces.

We apply these results to derive various inequalities for martingales in noncommutative symmetric quasi-Banach spaces.

Keywords

Noncommutative martingalesmartingale Hardy spacesinterpolation spaces

MSC classification

Primary: 46L53: Noncommutative probability and statistics 46B70: Interpolation between normed linear spaces 46L52: Noncommutative function spaces
Secondary: 46A13: Spaces defined by inductive or projective limits (LB, LF, etc.) 46A16: Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.) 60G50: Sums of independent random variables; random walks
Type
Article
Information
Canadian Journal of Mathematics , Volume 74 , Issue 6 , December 2022 , pp. 1700 - 1744
Check for updates
Copyright
© Canadian Mathematical Society 2021

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bekjan, T., Chen, Z., Perrin, M., and Yin, Z., Atomic decomposition and interpolation for Hardy spaces of noncommutative martingales . J. Funct. Anal. 258(2010), no. 7, 2483–2505.CrossRefGoogle Scholar
Bennett, C. and Sharpley, R., Interpolation of operators. Academic Press, Boston, MA, 1988.Google Scholar
Bergh, J. and Löfström, J., Interpolation spaces. An introduction. Vol. 223, Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 1976.CrossRefGoogle Scholar
Cadilhac, L., Noncommutative Khintchine inequalities in interpolation spaces of $\,{L}_p$ -spaces . Adv. Math. 352(2019), 265–296.CrossRefGoogle Scholar
Cadilhac, L., Majorization, interpolation and noncommutative Khinchin inequalities . Studia Math. 258(2021), no. 1, 1–26.CrossRefGoogle Scholar
Cadilhac, L. and Ricard, E., Sums of free variables in fully symmetric spaces. Proc. Lond. Math. Soc. (3), 122 (2021) No. 5, 724–744.Google Scholar
Chen, Z., Randrianantoanina, N., and Xu, Q., Atomic decompositions for noncommutative martingales. Preprint, 2020. arXiv:2001.08775v1 Google Scholar
Cuculescu, I., Martingales on von Neumann algebras . J. Multivariate Anal. 1(1971), 17–27.CrossRefGoogle Scholar
Cwikel, M., Milman, M., and Sagher, Y., Complex interpolation of some quasi-Banach spaces . J. Funct. Anal. 65(1986), no. 3, 339–347.CrossRefGoogle Scholar
Dirksen, S., Noncommutative and vector-valued Rosenthal inequalities. Ph.D. thesis dissertation, Netherlands Organisation for Scientific Research, 2011.Google Scholar
Dirksen, S., Noncommutative Boyd interpolation theorems . Trans. Amer. Math. Soc. 367(2015), no. 6, 4079–4110.CrossRefGoogle Scholar
Fack, T. and Kosaki, H., Generalized $s$ -numbers of $\,\tau$ -measurable operators . Pacific J. Math. 123(1986), 269–300.CrossRefGoogle Scholar
Fefferman, C. and Stein, E. M., ${H}^p$ spaces of several variables . Acta Math. 129(1972), no. 3–4, 137–193.CrossRefGoogle Scholar
Garsia, A. M., Martingale inequalities: seminar notes on recent progress, Mathematics Lecture Notes Series, W. A. Benjamin, Reading, MA, 1973.Google Scholar
Hitczenko, P. and Montgomery-Smith, S., Tangent sequences in Orlicz and rearrangement invariant spaces . Math. Proc. Cambridge Philos. Soc. 119(1996), no. 1, 91–101.CrossRefGoogle Scholar
Holmstedt, T., Interpolation of quasi-normed spaces . Math. Scand. 26(1970), 177–199.Google Scholar
Hong, G., Junge, M., and Parcet, J., Algebraic Davis decomposition and asymmetric Doob inequalities . Comm. Math. Phys. 346(2016), no. 3, 995–1019.CrossRefGoogle Scholar
Janson, S. and Jones, P. W., Interpolation between $\,{H}^p$ spaces: the complex method . J. Funct. Anal. 48(1982), no. 1, 58–80.CrossRefGoogle Scholar
Jiao, Y., Martingale inequalities in noncommutative symmetric spaces . Arch. Math. (Basel). 98(2012), no. 1, 87–97.CrossRefGoogle Scholar
Jiao, Y., Randrianantoanina, N., Wu, L., and Zhou, D., Square functions for noncommutative differentially subordinate martingales . Comm. Math. Phys. 374(2020), no. 2, 975–1019.CrossRefGoogle Scholar
Jiao, Y., Sukochev, F., and Zanin, D., Johnson–Schechtman and Khintchine inequalities in noncommutative probability theory . J. Lond. Math. Soc. 94(2016), no. 1, 113–140.CrossRefGoogle Scholar
Jiao, Y., Sukochev, F., Zanin, D., and Zhou, D., Johnson–Schechtman inequalities for noncommutative martingales . J. Funct. Anal. 272(2017), no. 3, 976–1016.CrossRefGoogle Scholar
Jones, P. W., On interpolation between ${H}^1$ and ${H}^{\infty }$ . In: Interpolation spaces and allied topics in analysis (Lund, 1983), Lecture Notes in Mathematics, 1070, Springer, Berlin, 1984, pp. 143–151.Google Scholar
Junge, M., Doob’s inequality for non-commutative martingales . J. Reine Angew. Math. 549(2002), 149–190.Google Scholar
Junge, M. and Perrin, M., Theory of ${\mathcal{\textbf{H}}}_p$ -spaces for continuous filtrations in von Neumann algebras . Astérisque. 362(2014), vi+134.Google Scholar
Junge, M. and Xu, Q., Noncommutative Burkholder/Rosenthal inequalities . Ann. Probab. 3(2003), no. 2, 948–995.Google Scholar
Kalton, N. and Montgomery-Smith, S., Interpolation of Banach spaces, Handbook of the geometry of Banach spaces. Vol. 2, North-Holland, Amsterdam, 2003, pp. 1131–1175.Google Scholar
Kalton, N. J. and Sukochev, F. A., Symmetric norms and spaces of operators . J. Reine Angew. Math. 621(2008), 81–121.Google Scholar
Kamińska, A., Maligranda, L., and Persson, L. E., Convexity, concavity, type and cotype of Lorentz spaces . Indag. Math. (N.S.). 9(1998), no. 3, 367–382.CrossRefGoogle Scholar
Kreĭn, S. G., Petunīn, Y. Ī., and Semënov, E. M., Interpolation of linear operators. Vol. 54. Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1982, Translated from the Russian by J. SzHucs.Google Scholar
Maligranda, L., Orlicz spaces and interpolation. Vol. 5, Seminários de Matemática [Seminars in Mathematics], Universidade Estadual de Campinas, Departamento de Matemática, Campinas, 1989.Google Scholar
Musat, M., Interpolation between non-commutative BMO and non-commutative $\,{L}_p$ -spaces . J. Funct. Anal. 202(2003), no. 1, 195–225.CrossRefGoogle Scholar
Perrin, M., A noncommutative Davis’ decomposition for martingales . J. Lond. Math. Soc. (2). 80(2009), no. 3, 627–648.CrossRefGoogle Scholar
Perrin, M., Inégalités de martingales non commutatives et applications. Ph.D. thesis dissertation, Université de Franche-Comté, 2011.Google Scholar
Pisier, G., Interpolation between ${H}^p$ spaces and noncommutative generalizations. I . Pacific J. Math. 155(1992), no. 2, 341–368.CrossRefGoogle Scholar
Pisier, G., Martingales in Banach spaces. Vol. 155, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2016.CrossRefGoogle Scholar
Pisier, G. and Xu, Q., Non-commutative martingale inequalities . Comm. Math. Phys. 189(1997), 667–698.CrossRefGoogle Scholar
Pisier, G. and Xu, Q., Non-commutative ${L}^p$ -spaces. In: Handbook of the geometry of Banach spaces, 2, North-Holland, Amsterdam, 2003, pp. 1459–1517.CrossRefGoogle Scholar
Qiu, Y., A non-commutative version of Lépingle–Yor martingale inequality . Statist. Probab. Lett. 91(2014), 52–54.CrossRefGoogle Scholar
Randrianantoanina, N., Square function inequalities for non-commutative martingales . Israel J. Math. 140(2004), 333–365.CrossRefGoogle Scholar
Randrianantoanina, N., Conditioned square functions for noncommutative martingales . Ann. Probab. 35(2007), no. 3, 1039–1070.CrossRefGoogle Scholar
Randrianantoanina, N. and Wu, L., Martingale inequalities in noncommutative symmetric spaces . J. Funct. Anal. 269(2015), no. 7, 2222–2253.CrossRefGoogle Scholar
Randrianantoanina, N., Wu, L., and Xu, Q., Noncommutative Davis type decompositions and applications . J. Lond. Math. Soc. (2). 99(2019), no. 1, 97–126.CrossRefGoogle Scholar
Randrianantoanina, N., Wu, L., and Zhou, D., Atomic decompositions and asymmetric Doob inequalities in noncommutative symmetric spaces . J. Funct. Anal. 280(2021), no. 1, 108794.CrossRefGoogle Scholar
Turpin, P., Convexités dans les espaces vectoriels topologiques généraux . Dissertationes Math. (Rozprawy Mat.). 131(1976), 221.Google Scholar
Weisz, F., Interpolation between martingale Hardy and BMO spaces, the real method . Bull. Sci. Math. 116(1992), no. 2, 145–158.Google Scholar
Weisz, F., Martingale Hardy spaces and their applications in Fourier analysis. Vol. 1568, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1994.CrossRefGoogle Scholar
Wolff, T. H., A note on interpolation spaces . In: Harmonic analysis (Minneapolis, MN, 1981). Vol. 908, Lecture Notes in Mathematics, Springer, Berlin, 1982, pp. 199–204.CrossRefGoogle Scholar
Xu, Q., Applications du théorème de factorisation pour des fonctions à valeurs opérateurs . Studia Math. 95(1990), no. 3, 273–292.CrossRefGoogle Scholar
Xu, Q., Analytic functions with values in lattices and symmetric spaces of measurable operators . Math. Proc. Cambridge Philos. Soc. 109(1991), 541–563.CrossRefGoogle Scholar