Burkholder’s inequalities in noncommutative Lorentz spaces

Jiao, Yong

doi:10.1090/S0002-9939-10-10267-6

Burkholder’s inequalities in noncommutative Lorentz spaces
HTML articles powered by AMS MathViewer

by Yong Jiao PDF

Proc. Amer. Math. Soc. 138 (2010), 2431-2441 Request permission

Abstract:

We prove Burkholder’s inequalities in noncommutative Lorentz spaces $L^{p,q}(\mathcal {M}), 1<p<\infty ,$ $1\leq q<\infty$, associated with a von Neumann algebra $\mathcal {M}$ equipped with a faithful normal tracial state. These estimates generalize the classical inequalities in the commutative case.

References

D. L. Burkholder, Distribution function inequalities for martingales, Ann. Probability 1 (1973), 19–42. MR 365692, DOI 10.1214/aop/1176997023
D. L. Burkholder and R. F. Gundy, Extrapolation and interpolation of quasi-linear operators on martingales, Acta Math. 124 (1970), 249–304. MR 440695, DOI 10.1007/BF02394573
Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 0482275, DOI 10.1007/978-3-642-66451-9
Colin Bennett and Robert Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Inc., Boston, MA, 1988. MR 928802
N. L. Carothers and S. J. Dilworth, Inequalities for sums of independent random variables, Proc. Amer. Math. Soc. 104 (1988), no. 1, 221–226. MR 958071, DOI 10.1090/S0002-9939-1988-0958071-4
Joseph Diestel, Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 1984. MR 737004, DOI 10.1007/978-1-4612-5200-9
Richard A. Hunt, On $L(p,\,q)$ spaces, Enseign. Math. (2) 12 (1966), 249–276. MR 223874
Richard O’Neil, Convolution operators and $L(p,\,q)$ spaces, Duke Math. J. 30 (1963), 129–142. MR 146673
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 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, Noncommutative Burkholder/Rosenthal inequalities. II. Applications, Israel J. Math. 167 (2008), 227–282. MR 2448025, DOI 10.1007/s11856-008-1048-4
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
Haskell P. Rosenthal, On the subspaces of $L^{p}$ $(p>2)$ spanned by sequences of independent random variables, Israel J. Math. 8 (1970), 273–303. MR 271721, DOI 10.1007/BF02771562
Narcisse Randrianantoanina, Conditioned square functions for noncommutative martingales, Ann. Probab. 35 (2007), no. 3, 1039–1070. MR 2319715, DOI 10.1214/009117906000000656
Q. Xu. Recent development on noncommutative martingale inequalities. Functional space theory and its applications, Proceedings of International Conference in China, Wuhan, 2003, 283-314.