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
. This and results of Bourgain are used to show that spaces
are not isomorphic to nonatomic Banach lattices. We also show that tent spaces introduced by Coifman, Meyer, and Stein are isomorphic to
.
- [1]
Ju. A. Abramovič and P. Wojtaszczyk, On the uniqueness of order in the spaces
and
, Mat. Zametki 18 (1975), 313-325. MR 0394097 (52:14903)
- [2]
J. Bourgain, Non-isomorphism of
-spaces in one and several variables, J. Funct. Anal. 46 (1982), 45-57. MR 654464 (84f:46073)
- [3]
-, The non-isomorphism of
-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
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
, 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)
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9939-1994-1181168-6
Keywords:
Order-continuous Banach lattice,
Hardy spaces
Article copyright:
© Copyright 1994
American Mathematical Society