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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Vector-valued inequalities for Fourier series


Author: José L. Rubio De Francia
Journal: Proc. Amer. Math. Soc. 78 (1980), 525-528
MSC: Primary 42A20
MathSciNet review: 556625
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Abstract: Denoting by $ {S^\ast}$ the maximal partial sum operator of Fourier series, we prove that $ {S^\ast}({f_1},{f_2}, \ldots ,{f_k}, \ldots ) = ({S^\ast}{f_1},{S^\ast}{f_2}, \ldots ,{S^\ast}{f_k}, \ldots )$ is a bounded operator from $ {L^p}({l^r})$ to itself, $ 1 < p,r < \infty $. Thus, we extend the theorem of Carleson and Hunt on pointwise convergence of Fourier series to the case of vector valued functions. We give also an application to the rectangular convergence of double Fourier series.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1980-0556625-3
PII: S 0002-9939(1980)0556625-3
Keywords: Convergence of Fourier series, maximal partial sum operator, vector valued Fourier series, double Fourier series
Article copyright: © Copyright 1980 American Mathematical Society