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The Littlewood—Paley—Rubio de Francia property of a Banach space for the case of equal intervals

Published online by Cambridge University Press:  08 July 2009

T. P. Hytönen ,
J. L. Torrea  and
D. V. Yakubovich
T. P. Hytönen
Affiliation:
Department of Mathematics and Statistics, University of Helsinki, Gustaf Hällströmin katu 2b, 00014 Helsinki, Finland ([email protected])
J. L. Torrea
Affiliation:
Departamento de Matemáticas, Universidad Autónoma de Madrid and ICMAT (CSIC–UAM–UC3M–UCM) Cantoblanco, 28049 Madrid, Spain ([email protected])
D. V. Yakubovich
Affiliation:
Departamento de Matemáticas, Universidad Autónoma de Madrid and ICMAT (CSIC–UAM–UC3M–UCM) Cantoblanco, 28049 Madrid, Spain ([email protected])
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Abstract

Let X be a Banach space. It is proved that an analogue of the Rubio de Francia square function estimate for partial sums of the Fourier series of X-valued functions holds true for all disjoint collections of subintervals of the set of integers of equal length and for all exponents p ≥ 2 if and only if the space X is a UMD space of type 2. The same criterion is obtained for the case of subintervals of the real line and Fourier integrals instead of Fourier series.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 139 , Issue 4 , August 2009 , pp. 819 - 832
Copyright
Copyright © Royal Society of Edinburgh 2009

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