Noncommutative maximal inequalities associated with convex functions

Bekjan, Turdebek; Chen, Zeqian; Osȩkowski, Adam

doi:10.1090/tran/6663

Noncommutative maximal inequalities associated with convex functions
HTML articles powered by AMS MathViewer

by Turdebek N. Bekjan, Zeqian Chen and Adam Osȩkowski PDF

Trans. Amer. Math. Soc. 369 (2017), 409-427 Request permission

Abstract:

We prove several noncommutative maximal inequalities associated with convex functions, including a Doob type inequality for a convex function of maximal operators on noncommutative martingales, and noncommutative Dunford-Schwartz and Stein maximal ergodic inequalities for a convex function of positive and symmetric positive contractions. The key ingredient in our proofs is a Marcinkiewicz type interpolation theorem for a convex function of maximal operators in the noncommutative setting, which we establish in this paper. These generalize the results of Junge and Xu in the $L^p$ case to the case of convex functions.

References

Charles A. Akemann, Joel Anderson, and Gert K. Pedersen, Triangle inequalities in operator algebras, Linear and Multilinear Algebra 11 (1982), no. 2, 167–178. MR 650729, DOI 10.1080/03081088208817440
Turdebek N. Bekjan, Noncommutative maximal ergodic theorems for positive contractions, J. Funct. Anal. 254 (2008), no. 9, 2401–2418. MR 2409167, DOI 10.1016/j.jfa.2008.01.007
Turdebek N. Bekjan and Zeqian Chen, Interpolation and $\Phi$-moment inequalities of noncommutative martingales, Probab. Theory Related Fields 152 (2012), no. 1-2, 179–206. MR 2875756, DOI 10.1007/s00440-010-0319-2
Turdebek N. Bekjan, Zeqian Chen, Peide Liu, and Yong Jiao, Noncommutative weak Orlicz spaces and martingale inequalities, Studia Math. 204 (2011), no. 3, 195–212. MR 2812201, DOI 10.4064/sm204-3-1
Turdebek N. Bekjan, Zeqian Chen, Mathilde Perrin, and Zhi Yin, Atomic decomposition and interpolation for Hardy spaces of noncommutative martingales, J. Funct. Anal. 258 (2010), no. 7, 2483–2505. MR 2584751, DOI 10.1016/j.jfa.2009.12.006
D. L. Burkholder, Distribution function inequalities for martingales, Ann. Probability 1 (1973), 19–42. MR 365692, DOI 10.1214/aop/1176997023
D. L. Burkholder, B. J. Davis, and R. F. Gundy, Integral inequalities for convex functions of operators on martingales, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971) Univ. California Press, Berkeley, Calif., 1972, pp. 223–240. MR 0400380
Zeqian Chen, Quanhua Xu, and Zhi Yin, Harmonic analysis on quantum tori, Comm. Math. Phys. 322 (2013), no. 3, 755–805. MR 3079331, DOI 10.1007/s00220-013-1745-7
I. Cuculescu, Martingales on von Neumann algebras, J. Multivariate Anal. 1 (1971), no. 1, 17–27. MR 295398, DOI 10.1016/0047-259X(71)90027-3
Burgess Davis, On the integrability of the martingale square function, Israel J. Math. 8 (1970), 187–190. MR 268966, DOI 10.1007/BF02771313
Andreas Defant and Marius Junge, Maximal theorems of Menchoff-Rademacher type in non-commutative $L_q$-spaces, J. Funct. Anal. 206 (2004), no. 2, 322–355. MR 2021850, DOI 10.1016/j.jfa.2002.07.001
Sjoerd Dirksen, Noncommutative Boyd interpolation theorems, Trans. Amer. Math. Soc. 367 (2015), no. 6, 4079–4110. MR 3324921, DOI 10.1090/S0002-9947-2014-06185-0
Sjoerd Dirksen, Weak-type interpolation for noncommutative maximal operators, J. Operator Theory 73 (2015), no. 2, 515–532. MR 3346135, DOI 10.7900/jot.2014mar12.2022
Sjoerd Dirksen and Éric Ricard, Some remarks on noncommutative Khintchine inequalities, Bull. Lond. Math. Soc. 45 (2013), no. 3, 618–624. MR 3065031, DOI 10.1112/blms/bds107
Peter G. Dodds, Theresa K.-Y. Dodds, and Ben de Pagter, Noncommutative Banach function spaces, Math. Z. 201 (1989), no. 4, 583–597. MR 1004176, DOI 10.1007/BF01215160
Peter G. Dodds, Theresa K. Dodds, and Ben de Pagter, Fully symmetric operator spaces, Integral Equations Operator Theory 15 (1992), no. 6, 942–972. MR 1188788, DOI 10.1007/BF01203122
Thierry Fack and Hideki Kosaki, Generalized $s$-numbers of $\tau$-measurable operators, Pacific J. Math. 123 (1986), no. 2, 269–300. MR 840845
Edward G. Effros and Zhong-Jin Ruan, Operator spaces, London Mathematical Society Monographs. New Series, vol. 23, The Clarendon Press, Oxford University Press, New York, 2000. MR 1793753
Fumio Hiai and Dénes Petz, The semicircle law, free random variables and entropy, Mathematical Surveys and Monographs, vol. 77, American Mathematical Society, Providence, RI, 2000. MR 1746976, DOI 10.1090/surv/077
Marius Junge, Doob’s inequality for non-commutative martingales, J. Reine Angew. Math. 549 (2002), 149–190. MR 1916654, DOI 10.1515/crll.2002.061
Marius Junge, Embedding of the operator space $OH$ and the logarithmic ‘little Grothendieck inequality’, Invent. Math. 161 (2005), no. 2, 225–286. MR 2180450, DOI 10.1007/s00222-004-0421-0
Marius Junge, Christian Le Merdy, and Quanhua Xu, $H^\infty$ functional calculus and square functions on noncommutative $L^p$-spaces, Astérisque 305 (2006), vi+138 (English, with English and French summaries). MR 2265255
Marius Junge and Javier Parcet, Operator space embedding of Schatten $p$-classes into von Neumann algebra preduals, Geom. Funct. Anal. 18 (2008), no. 2, 522–551. MR 2421547, DOI 10.1007/s00039-008-0660-0
Marius Junge and Quanhua Xu, Noncommutative Burkholder/Rosenthal inequalities, Ann. Probab. 31 (2003), no. 2, 948–995. MR 1964955, DOI 10.1214/aop/1048516542
Marius Junge and Quanhua Xu, On the best constants in some non-commutative martingale inequalities, Bull. London Math. Soc. 37 (2005), no. 2, 243–253. MR 2119024, DOI 10.1112/S0024609304003947
Marius Junge and Quanhua Xu, Noncommutative maximal ergodic theorems, J. Amer. Math. Soc. 20 (2007), no. 2, 385–439. MR 2276775, DOI 10.1090/S0894-0347-06-00533-9
Marius Junge and Quanhua Xu, Noncommutative Burkholder/Rosenthal inequalities. II. Applications, Israel J. Math. 167 (2008), 227–282. MR 2448025, DOI 10.1007/s11856-008-1048-4
Marius Junge and Quanhua Xu, Representation of certain homogeneous Hilbertian operator spaces and applications, Invent. Math. 179 (2010), no. 1, 75–118. MR 2563760, DOI 10.1007/s00222-009-0210-x
N. J. Kalton and F. A. Sukochev, Symmetric norms and spaces of operators, J. Reine Angew. Math. 621 (2008), 81–121. MR 2431251, DOI 10.1515/CRELLE.2008.059
Christian Le Merdy and Quanhua Xu, Maximal theorems and square functions for analytic operators on $L^p$-spaces, J. Lond. Math. Soc. (2) 86 (2012), no. 2, 343–365. MR 2980915, DOI 10.1112/jlms/jds009
Christian Le Merdy and Quanhua Xu, Strong $q$-variation inequalities for analytic semigroups, Ann. Inst. Fourier (Grenoble) 62 (2012), no. 6, 2069–2097 (2013) (English, with English and French summaries). MR 3060752, DOI 10.5802/aif.2743
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. Function spaces. MR 540367
Lech Maligranda, Indices and interpolation, Dissertationes Math. (Rozprawy Mat.) 234 (1985), 49. MR 820076
Lech Maligranda, Orlicz spaces and interpolation, Seminários de Matemática [Seminars in Mathematics], vol. 5, Universidade Estadual de Campinas, Departamento de Matemática, Campinas, 1989. MR 2264389
Javier Parcet and Narcisse Randrianantoanina, Gundy’s decomposition for non-commutative martingales and applications, Proc. London Math. Soc. (3) 93 (2006), no. 1, 227–252. MR 2235948, DOI 10.1017/S0024611506015863
Mathilde Perrin, A noncommutative Davis’ decomposition for martingales, J. Lond. Math. Soc. (2) 80 (2009), no. 3, 627–648. MR 2559120, DOI 10.1112/jlms/jdp045
Gilles Pisier, Non-commutative vector valued $L_p$-spaces and completely $p$-summing maps, Astérisque 247 (1998), vi+131 (English, with English and French summaries). MR 1648908
Gilles Pisier, Introduction to operator space theory, London Mathematical Society Lecture Note Series, vol. 294, Cambridge University Press, Cambridge, 2003. MR 2006539, DOI 10.1017/CBO9781107360235
Gilles Pisier and Dimitri Shlyakhtenko, Grothendieck’s theorem for operator spaces, Invent. Math. 150 (2002), no. 1, 185–217. MR 1930886, DOI 10.1007/s00222-002-0235-x
Gilles Pisier and Quanhua Xu, Non-commutative martingale inequalities, Comm. Math. Phys. 189 (1997), no. 3, 667–698. MR 1482934, DOI 10.1007/s002200050224
Gilles Pisier and Quanhua Xu, Non-commutative $L^p$-spaces, Handbook of the geometry of Banach spaces, Vol. 2, North-Holland, Amsterdam, 2003, pp. 1459–1517. MR 1999201, DOI 10.1016/S1874-5849(03)80041-4
Narcisse Randrianantoanina, Non-commutative martingale transforms, J. Funct. Anal. 194 (2002), no. 1, 181–212. MR 1929141
Narcisse Randrianantoanina, Conditioned square functions for noncommutative martingales, Ann. Probab. 35 (2007), no. 3, 1039–1070. MR 2319715, DOI 10.1214/009117906000000656
E. Ricard and Q. Xu, A noncommutative martingale convexity inequality, arXiv:1405.0431.
D. V. Voiculescu, K. J. Dykema, and A. Nica, Free random variables, CRM Monograph Series, vol. 1, American Mathematical Society, Providence, RI, 1992. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups. MR 1217253, DOI 10.1090/crmm/001
Quan Hua Xu, Analytic functions with values in lattices and symmetric spaces of measurable operators, Math. Proc. Cambridge Philos. Soc. 109 (1991), no. 3, 541–563. MR 1094753, DOI 10.1017/S030500410006998X
Quanhua Xu, Operator-space Grothendieck inequalities for noncommutative $L_p$-spaces, Duke Math. J. 131 (2006), no. 3, 525–574. MR 2219250, DOI 10.1215/S0012-7094-06-13135-6
Q. Xu, Noncommutative $L_p$-Spaces and Martingale Inequalities, book manuscript, 2007.
F. J. Yeadon, Ergodic theorems for semifinite von Neumann algebras. I, J. London Math. Soc. (2) 16 (1977), no. 2, 326–332. MR 487482, DOI 10.1112/jlms/s2-16.2.326