Abstract
We prove a Rosenthal-type inequality for some sequences of independent random variables in BMO. This leads to (isomorphically) new complemented subspaces of H 1(δ), one of which is also translation invariant.
This inequality is used also to show that a complemented subspace of H 1 (resp. VMO) either contains a copy of l 2 or is isomorphic to a complemented subspace of (Σ H 1n )1 (resp. (ΣBMO n)co). We thus verify a conjecture of P. Wojtaszczyk.
We also show that Hilbertian subspaces of VMO are complemented, and that the Walsh functions of multiplicity k, k≥2, span uncomplemented copies of l 2 in H 1.
Supported by Erwin Schrödinger-Auslandsstipendium Pr. Nr. J0288P.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
J. Bourgain, H. Rosenthal and G. Schechtman, An ordinal L p-index for Banach spaces, with application to complemented subspaces of L p, Ann. Math. 114 (1981), 193–228.
A. Garsia, Martingale inequalities, W.A. Benjamin, 1973.
W.B. Johnson, Operators into L p, which factor through l p, J. of London Math. Soc. 14 (1976), 333–339.
I. Klemes, The idempotent multipliers of H 1(T), Can. J. Math. 39 (1987), 1223–1234.
S. Kwapien and A. Pelczynski, Some linear topological properties of the Hardy-Spaces H p, Compositio Math. 33 (1976), 261–288.
P.F.X. Müller, Classification of the isomorphic types of Martingale H 1 spaces, Israel J. of Math. 59 (1987), 195–211.
J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Springer Verlag, 1977.
J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II, Springer Verlag, 1979.
H.P. Rosenthal, On the subspaces of L p (p > 2) spanned by sequences of independent random variables, Israel J. of Math. 8 (1970), 273–303.
G. Schechtman, Examples of \(\mathcal{L}_p\) spaces (1<p ≠ 2 < ∞), Israel J. of Math. 22 (1975), 138–147.
P. Wojtaszczyk, The Banach Space H 1, in Functional Analysis: Surveys and recent results III (K.D. Bierstedt, B. Fuchssteiner, eds.) North-Holland 1984.
J.Y.T. Woo, On the class of universal modular sequence spaces, Israel J. Math. 20 (1975), 193–215.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1989 Springer-Verlag
About this chapter
Cite this chapter
Müller, P.F.X., Schechtman, G. (1989). On complemented subspaces of H 1 and VMO . In: Lindenstrauss, J., Milman, V.D. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 1376. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0090051
Download citation
DOI: https://doi.org/10.1007/BFb0090051
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-51303-2
Online ISBN: 978-3-540-46189-0
eBook Packages: Springer Book Archive