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
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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. Voĭtaščik, The uniqueness of the order in the spaces 𝐿_{𝑝}[0,1] and 1_{𝑝}, Mat. Zametki 18 (1975), no. 3, 313–325 (Russian). MR 0394097
[2] J. Bourgain, The nonisomorphism of 𝐻¹-spaces in one and several variables, J. Funct. Anal. 46 (1982), no. 1, 45–57. MR 654464, https://doi.org/10.1016/0022-1236(82)90043-X
[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. M. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal. 62 (1985), no. 2, 304–335. MR 791851, https://doi.org/10.1016/0022-1236(85)90007-2
[5] T. Figiel, An example of infinite dimensional reflexive Banach space non-isomorphic to its Cartesian square, Studia Math. 42 (1972), 295–306. MR 0306875
[6] W. T. Gowers, A solution to Banach's hyperplane problem (to appear).
[7] Eleonor Harboure, José L. Torrea, and Beatriz E. Viviani, A vector-valued approach to tent spaces, J. Analyse Math. 56 (1991), 125–140. MR 1243101, https://doi.org/10.1007/BF02820462
[8] William B. Johnson, A complementary universal conjugate Banach space and its relation to the approximation problem, Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 1972), 1972, pp. 301–310 (1973). MR 0326356, https://doi.org/10.1007/BF02762804
[9] W. B. Johnson, B. Maurey, G. Schechtman, and L. Tzafriri, Symmetric structures in Banach spaces, Mem. Amer. Math. Soc. 19 (1979), no. 217, v+298. MR 527010, https://doi.org/10.1090/memo/0217
[10] N. J. Kalton, Lattice structures on Banach spaces, Mem. Amer. Math. Soc. 103 (1993), no. 493, vi+92. MR 1145663, https://doi.org/10.1090/memo/0493
[11] N. J. Kalton, C. Leránoz, and P. Wojtaszczyk, Uniqueness of unconditional bases in quasi-Banach spaces with applications to Hardy spaces, Israel J. Math. 72 (1990), no. 3, 299–311 (1991). MR 1120223, https://doi.org/10.1007/BF02773786
[12] N. J. Kalton, N. T. Peck, and James W. Roberts, An 𝐹-space sampler, London Mathematical Society Lecture Note Series, vol. 89, Cambridge University Press, Cambridge, 1984. MR 808777
[13] J. L. Krivine, Théorèmes de factorisation dans les espaces réticulés, Séminaire Maurey-Schwartz 1973–1974: Espaces 𝐿^{𝑝}, applications radonifiantes et géométrie des espaces de Banach, Exp. Nos. 22 et 23, Centre de Math., École Polytech., Paris, 1974, pp. 22 (French). MR 0440334
[14] E. Lacey and P. Wojtaszczyk, Nonatomic Banach lattices can have 𝑙₁ as a dual space, Proc. Amer. Math. Soc. 57 (1976), no. 1, 79–84. MR 0402459, https://doi.org/10.1090/S0002-9939-1976-0402459-9
[15] Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Springer-Verlag, Berlin-New York, 1977. Sequence spaces; Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92. MR 0500056
[16] 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
[17] Bernard Maurey, Isomorphismes entre espaces 𝐻₁, Acta Math. 145 (1980), no. 1-2, 79–120 (French). MR 586594, https://doi.org/10.1007/BF02414186
[18] Walter Rudin, Function theory in polydiscs, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0255841
[19] Joel H. Shapiro, Mackey topologies, reproducing kernels, and diagonal maps on the Hardy and Bergman spaces, Duke Math. J. 43 (1976), no. 1, 187–202. MR 0500100
[20] Per Sjölin and Jan-Olov Strömberg, Basis properties of Hardy spaces, Ark. Mat. 21 (1983), no. 1, 111–125. MR 706642, https://doi.org/10.1007/BF02384303
[21] Alberto Torchinsky, Real-variable methods in harmonic analysis, Pure and Applied Mathematics, vol. 123, Academic Press, Inc., Orlando, FL, 1986. MR 869816
[22] P. Wojtaszczyk, Banach spaces for analysts, Cambridge Studies in Advanced Mathematics, vol. 25, Cambridge University Press, Cambridge, 1991. MR 1144277