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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On rearrangements of Vilenkin-Fourier series which preserve almost everywhere convergence

Authors: J. A. Gosselin and W. S. Young
Journal: Trans. Amer. Math. Soc. 209 (1975), 157-174
MSC: Primary 43A50; Secondary 42A56
MathSciNet review: 0399756
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Abstract: It is known that the partial sums of Vilenkin-Fourier series of ${L^q}$ functions $(q > 1)$ converge a.e. In this paper we establish the ${L^2}$ result for a class of rearrangements of the Vilenkin-Fourier series, and the ${L^q}$ result $(1 < q < 2)$ for a subclass of rearrangements. In the case of the Walsh-Fourier series, these classes include the Kaczmarz rearrangement studied by L. A. Balashov. The ${L^2}$ result for the Kaczmarz rearrangement was first proved by K. H. Moon. The techniques of proof involve a modification of the Carleson-Hunt method and estimates on maximal functions of the Hardy-Littlewood type that arise from these rearrangements.

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Keywords: Hardy-Littlewood maximal function, maximal <IMG WIDTH="18" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$n$">th partial sum operator, conditional expectation, weak type inequalities
Article copyright: © Copyright 1975 American Mathematical Society