Boundedness of the q-Mean-Square Operator on Vector-Valued Analytic Martingales

Peide, Liu ; Saksman, Eero ; Tylli, Hans-Olav

Canadian mathematical bulletin, Tome 42 (1999) no. 2, pp. 221-230

Résumé

We study boundedness properties of the $q$ -mean-square operator {{S}^{(q)}} on $E$ -valued analytic martingales, where $E$ is a complex quasi-Banach space and $2\,\le \,q\,<\,\infty $ . We establish that a.s. finiteness of ${{S}^{(q)}}$ for every bounded $E$ -valued analytic martingale implies strong $(p,\,p)$ -type estimates for ${{S}^{(q)}}$ and all $p\,\in \,(0,\,\infty )$ . Our results yield new characterizations (in terms of analytic and stochastic properties of the function ${{S}^{(q)}}$ ) of the complex spaces $E$ that admit an equivalent $q$ -uniformly $\text{PL}$ -convex quasi-norm. We also obtain a vector-valued extension (and a characterization) of part of an observation due to Bourgain and Davis concerning the ${{L}^{p}}$ -boundedness of the usual square-function on scalar-valued analytic martingales.

Détail

DOI : 10.4153/CMB-1999-027-3

Mots-clés : 46B20, 60G46

Bibliographie

[A] [A] Aleksandrov, A. B., Essays on non locally convex Hardy classes. Lecture Notes in Math. 864, Springer-Verlag, 1981, 1–89. Google Scholar

[BD] [BD] Bourgain, J. and Davis, W. J., Martingale transforms and complex uniform convexity. Trans. Amer.Math. Soc. 294 (1986), 501–515. Google Scholar

[B1] [B1] Burkholder, D. L., Distribution function inequalities for martingales. Ann. Probab. 1 (1973), 19–42. Google Scholar

[B2] [B2] Burkholder, D. L., A geometric characterization of Banach spaces in which martingale difference sequences are unconditional. Ann. Probab. 9 (1981), 991–1011. Google Scholar

[B3] [B3] Burkholder, D. L.,Martingale transforms and the geometry of Banach spaces. Lecture Notes in Math. 860, Springer-Verlag, 1981, 35–50. Google Scholar

[DGT] [DGT] Davis, W. J., Garling, D. J. H. and Tomczak-Jaegermann, N., The complex convexity of quasi-normed linear spaces. J. Funct. Anal. 55 (1984), 110–150. Google Scholar

[DU] [DU] Diestel, J. and Uhl, J. J. Jr., Vector measures. Mathematical Surveys 15, Amer. Math. Soc., Providence, RI, 1977. Google Scholar

[D] [D] Dilworth, S. J., Complex convexity and the geometry of Banach spaces.Math. Proc. Cambridge Philos. Soc. 99 (1986), 495–506. Google Scholar

[Ed1] [Ed1] Edgar, G. A., Complex martingale convergence. Lecture Notes in Math. 1166, Springer-Verlag, 1985, 38–59. Google Scholar

[Ed2] [Ed2] Edgar, G. A., Analytic martingale convergence. J. Funct. Anal. 69 (1986), 268–280. Google Scholar

[E] [E] Enflo, P., Banach spaces which can be given an equivalent uniformly convex norm. Israel J. Math. 13 (1972), 281–288. Google Scholar

[G1] [G1] Garling, D. J. H., On martingales with values in a complex Banach space. Math. Proc. Cambridge Philos. Soc. 104 (1988), 399–406. Google Scholar

[G2] [G2] Garling, D. J. H., On the dual of a proper uniform algebra. Bull. LondonMath. Soc. 21 (1989), 279–284. Google Scholar

[Ge] [Ge] Geiss, S., BMO -spaces and applications to extrapolation theory. Studia Math. 122 (1997), 235–274. Google Scholar

[HP] [HP] Haagerup, U. and Pisier, G., Factorization of analytic functions with values in non-commutative L1-spaces and applications. Canad. J. Math. 41 (1989), 882–906. Google Scholar

[K] [K] König, H., Eigenvalue distributions of compact operators. Birkhäuser, 1986. Google Scholar

[L] [L] Long, Ruilin, Martingale spaces and inequalities. Vieweg Verlag and Peking University Press, 1993. Google Scholar

[M] [M] Mattila, K., Complex strict and uniform convexity and hyponormal operators. Math. Proc. Cambridge Philos. Soc. 96 (1984), 483–493. Google Scholar

[P] [P] Pisier, G., Martingales with values in uniformly convex spaces. Israel J. Math. 20 (1975), 326–350. Google Scholar

[X1] [X1] Xu, Quanhua, Inégalités pour les martingales de Hardy et renormage des espaces quasinormés. C. R. Acad. Sci. Paris 306 (1988), 601–604. Google Scholar

[X2] [X2] Xu, Quanhua, Convexités uniformes et inégalités de martingales. Math. Ann. 287 (1990), 193–211. Google Scholar