On nonatomic Banach lattices and Hardy spaces

Kalton, N. J.; Wojtaszczyk, P.

doi:10.1090/S0002-9939-1994-1181168-6

On nonatomic Banach lattices and Hardy spaces

Authors: N. J. Kalton and P. Wojtaszczyk
Journal: Proc. Amer. Math. Soc. 120 (1994), 731-741
MSC: Primary 46B42; Secondary 42B30, 46E15
DOI: https://doi.org/10.1090/S0002-9939-1994-1181168-6
MathSciNet review: 1181168
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We are interested in the question when a Banach space with an unconditional basis is isomorphic (as a Banach space) to an order-continuous nonatomic Banach lattice. We show that this is the case if and only if is isomorphic as a Banach space with $X({\ell _2})$ . This and results of Bourgain are used to show that spaces ${H_1}({{\mathbf{T}}^n})$ are not isomorphic to nonatomic Banach lattices. We also show that tent spaces introduced by Coifman, Meyer, and Stein are isomorphic to $\operatorname{Rad} \;{H_1}$ .

[1] Ju. A. Abramovič and P. Wojtaszczyk, On the uniqueness of order in the spaces ${\ell _p}$ and ${L_p}[0,1]$ , Mat. Zametki 18 (1975), 313-325. MR 0394097 (52:14903)
[2] J. Bourgain, Non-isomorphism of ${H^1}$ -spaces in one and several variables, J. Funct. Anal. 46 (1982), 45-57. MR 654464 (84f:46073)
[3] -, The non-isomorphism of ${H^1}$ -spaces in different number of variables, Bull. Soc. Math. Belg. Ser. B 35 (1983), 127-136.
[4] R. R. Coifman, Y. Meyer, and E. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal. 62 (1985), 304-335. MR 791851 (86i:46029)
[5] T. Figiel, An example of an infinite dimensional, reflexive Banach space non-isomorphic to its Cartesian square, Studia Math. 42 (1972), 295-306. MR 0306875 (46:5997)
[6] W. T. Gowers, A solution to Banach's hyperplane problem (to appear).
[7] E. Harboure, J. L. Torrea and B. E. Viviani, A vector valued approach to tent spaces, J. Analyse Math. 66 (1991), 125-140. MR 1243101 (94i:42019)
[8] W. B. Johnson, A complementably universal conjugate Banach space and its relation to the approximation property, Israel J. Math. 13 (1972), 301-310. MR 0326356 (48:4700)
[9] W. B. Johnson, B. Maurey, G. Schechtman, and L. Tzafriri, Symmetric structures in Banach spaces, Mem. Amer. Math. Soc., No. 217, Amer. Math. Soc., Providence, RI, 1979. MR 527010 (82j:46025)
[10] N. J. Kalton, Lattice structures on Banach spaces, Mem. Amer. Math. Soc., no. 493, vol. 103, Amer. Math. Soc., Providence, RI, 1993. MR 1145663 (93j:46024)
[11] N. J. Kalton, C. Leranoz, and P. Wojtaszczyk, Uniqueness of unconditional bases in quasi-Banach spaces with applications to Hardy spaces, Israel J. Math. 72 (1990), 299-311. MR 1120223 (92f:46004)
[12] N. J. Kalton, N. T. Peck, and J. W. Roberts, An -space sampler, London Math. Soc. Lecture Note Ser., vol. 89, Cambridge Univ. Press, Cambridge, 1984. MR 808777 (87c:46002)
[13] J. L. Krivine, Théorèmes de factorisation dans les espaces réticules, Seminaire Maurey-Schwartz 1973-74, Exposes 22-23, École Polytechnique, Paris. MR 0440334 (55:13209)
[14] E. Lacey and P. Wojtaszczyk, Nonatomic Banach lattices can have ${\ell _1}$ as a dual space, Proc. Amer. Math. Soc. 57 (1976), 79-84. MR 0402459 (53:6279)
[15] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces. I. Sequence spaces, Springer-Verlag, Berlin, 1977. MR 0500056 (58:17766)
[16] -, Classical Banach spaces. II. Function spaces, Springer-Verlag, Berlin, 1979. MR 540367 (81c:46001)
[17] B. Maurey, Isomorphismes entre espaces ${H_1}$ , Acta Math. 145 (1980), 79-120. MR 586594 (84b:46027)
[18] W. Rudin, Function theory in polydiscs, Benjamin, New York, 1969. MR 0255841 (41:501)
[19] J. H. Shapiro, Mackey topologies, reproducing kernels, and diagonal maps on the Hardy and Bergman spaces, Duke Math. J. 43.1 (1976), 187-202. MR 0500100 (58:17806)
[20] P. Sjölin and J.-O. Stromberg, Basis properties of Hardy spaces, Ark. Mat. 21 (1983), 111-125. MR 706642 (85a:42037)
[21] A. Torchinsky, Real-variable methods in harmonic analysis, Academic Press, New York, 1986. MR 869816 (88e:42001)
[22] P. Wojtaszczyk, Banach spaces for analysts, Cambridge Stud. Adv. Math., vol. 25, Cambridge Univ. Press, Cambridge, 1991. MR 1144277 (93d:46001)