×

Uniqueness of unconditional bases in quasi-Banach spaces with applications to Hardy spaces. (English) Zbl 0753.46013

Summary: We prove some general results on the uniqueness of unconditional bases in quasi-Banach spaces. We show in particular that certain Lorentz spaces have unique unconditional bases answering a question of Nawrocki and Ortynski. We then given applications of these results to Hardy spaces by showing the spaces \(H_ p({\mathbf T}^ n)\) are mutually non-isomorphic for differing values of \(n\) when \(0<p<1\).

MSC:

46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] A. Bonami,Ensembles {\(\Lambda\)}(p)dans le dual D Ann. Inst. Fourier (Grenoble)18 (2) (1968), 193–204.
[2] J. Bourgain,The non-isomorphism of H 1-spaces in one and several variables, J. Funct. Anal.46 (1982), 45–57. · Zbl 0492.46043
[3] J. Bourgain,The non-isomorphism of H 1-spaces in a different number of variables, Bull. Soc. Math. Belg. Ser. B35 (1983), 127–136. · Zbl 0533.46036
[4] J. Bourgain, P. G. Casazza, J. Lindenstrauss and L. Tzafriri,Banach spaces with a unique unconditional basis, up to a permutation, Memoirs Am. Math. Soc. No. 322, Providence, 1985. · Zbl 0575.46011
[5] R. R. Coifman and R. Rochberg,Representation theorems for holomorphic and harmonic functions in L p , Asterisque77 (1980), 11–66. · Zbl 0472.46040
[6] N. J. Kalton,Orlicz sequence spaces without local convexity, Math. Proc. Camb. Phil. Soc.81 (1977), 253–278. · Zbl 0345.46013
[7] N. J. Kalton,Convexity conditions on non-locally convex lattices, Glasgow Math. J.25 (1984), 141–152. · Zbl 0564.46004
[8] N. J. Kalton and D. A. Trautman,Remarks on subspaces of H p when 0<p<1, Michigan Math. J.29 (1982), 163–170. · Zbl 0516.46038
[9] C. Leranoz, Ph.D. thesis, University of Missouri-Columbia, in preparation.
[10] J. Lindenstrauss and A. Pełczyński, Absolutely summing operators in -spaces and their applications, Studia Math.29 (1968), 275–326. · Zbl 0183.40501
[11] J. Lindenstrauss and L. Tzafriri,Classical Banach Spaces II, Function-Spaces, Springer-Verlag, Berlin-Heidelberg-New York, 1979. · Zbl 0403.46022
[12] J. Lindenstrauss and M. Zippin,Banach spaces with a unique unconditional basis, J. Funct. Anal.3 (1969), 115–125. · Zbl 0174.17201
[13] B. Maurey,Type et cotype dans les espaces munis de structures locales inconditionelles, Seminaire Maurey-Schwartz 1973–74, Exposes 24–25, Ecole Polytechnique, Paris.
[14] B. Maurey,Isomorphisms entre espaces H 1, Acta Math.145 (1980), 79–120. · Zbl 0509.46045
[15] M. Nawrocki,The non-isomorphism of the Smirnov classes of different balls and polydiscs, Bull. Soc. Math. Belg. Ser. B41 (1989), 307–315. · Zbl 0704.46012
[16] M. Nawrocki and A. Ortynski,The Mackey topology and complemented subspaces of Lorentz sequence spaces d(w, p) for 0<p<1, Trans. Am. Math. Soc.287 (1985), 713–722. · Zbl 0537.46012
[17] N. Popa,Basic sequences and subspaces in Lorentz sequence spaces without local convexity, Trans. Am. Math. Soc.263 (1981), 431–456. · Zbl 0461.46006
[18] H. P. Rosenthal and S. J. Szarek,On tensor products of operators from L p to L q , to appear. · Zbl 0758.47022
[19] P. Wojtaszczyk,H p -spaces, p,and spline systems, Studia Math.77 (1984), 289–320. · Zbl 0546.30026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.