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
HTML articles powered by AMS MathViewer

by N. J. Kalton and P. Wojtaszczyk PDF

Proc. Amer. Math. Soc. 120 (1994), 731-741 Request permission

Abstract:

We are interested in the question when a Banach space $X$ 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 $X$ 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 $\operatorname {Rad} \;{H_1}$.

References

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 394097
J. Bourgain, The nonisomorphism of $H^{1}$-spaces in one and several variables, J. Functional Analysis 46 (1982), no. 1, 45–57. MR 654464, DOI 10.1016/0022-1236(82)90043-X

The non-isomorphism of

spaces in different number of variables

35

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, DOI 10.1016/0022-1236(85)90007-2
T. Figiel, An example of infinite dimensional reflexive Banach space non-isomorphic to its Cartesian square, Studia Math. 42 (1972), 295–306. MR 306875, DOI 10.4064/sm-42-3-295-306

A solution to Banach’s hyperplane problem

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, DOI 10.1007/BF02820462
William B. Johnson, A complementary universal conjugate Banach space and its relation to the approximation problem, Israel J. Math. 13 (1972), 301–310 (1973). MR 326356, DOI 10.1007/BF02762804
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, DOI 10.1090/memo/0217
N. J. Kalton, Lattice structures on Banach spaces, Mem. Amer. Math. Soc. 103 (1993), no. 493, vi+92. MR 1145663, DOI 10.1090/memo/0493
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, DOI 10.1007/BF02773786
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, DOI 10.1017/CBO9780511662447
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
E. Lacey and P. Wojtaszczyk, Nonatomic Banach lattices can have $l_1$ as a dual space, Proc. Amer. Math. Soc. 57 (1976), no. 1, 79–84. MR 402459, DOI 10.1090/S0002-9939-1976-0402459-9
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056
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
Bernard Maurey, Isomorphismes entre espaces $H_{1}$, Acta Math. 145 (1980), no. 1-2, 79–120 (French). MR 586594, DOI 10.1007/BF02414186
Walter Rudin, Function theory in polydiscs, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0255841
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 500100
Per Sjölin and Jan-Olov Strömberg, Basis properties of Hardy spaces, Ark. Mat. 21 (1983), no. 1, 111–125. MR 706642, DOI 10.1007/BF02384303
Alberto Torchinsky, Real-variable methods in harmonic analysis, Pure and Applied Mathematics, vol. 123, Academic Press, Inc., Orlando, FL, 1986. MR 869816
P. Wojtaszczyk, Banach spaces for analysts, Cambridge Studies in Advanced Mathematics, vol. 25, Cambridge University Press, Cambridge, 1991. MR 1144277, DOI 10.1017/CBO9780511608735