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Pointwise convergence of lacunary partial sums of almost periodic Fourier series

Author: Andrew D. Bailey
Journal: Proc. Amer. Math. Soc. 142 (2014), 1757-1771
MSC (2012): Primary 42A75; Secondary 42A24, 42B25
Published electronically: February 19, 2014
MathSciNet review: 3168481
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Abstract: It is a classical result that lacunary partial sums of the Fourier series of functions $ f \in L^p(\mathbb{T})$ converge almost everywhere for $ p \in (1, \infty )$. In 1968, E. A. Bredihina established an analogous result for functions belonging to the Stepanov space of almost periodic functions $ S^2$ whose Fourier exponents satisfy a natural separation condition. Here, the maximal operators corresponding to lacunary partial summation of almost periodic Fourier series are shown to be bounded on the Stepanov spaces $ S^{2^k}$, $ k \in \mathbb{N}$, for functions satisfying the same condition; Bredihina's result follows as a consequence. In the process of establishing these bounds, some general results are obtained which will facilitate further work on operator bounds and convergence issues in Stepanov spaces. These include a boundedness theorem for the Hilbert transform and a theorem of Littlewood-Paley type. An improvement of ``$ S^{2^k}$, $ k \in \mathbb{N}$'' to ``$ S^p$, $ p \in (1, \infty )$'' is also seen to follow from a natural conjecture on the boundedness of the Hilbert transform.

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Additional Information

Andrew D. Bailey
Affiliation: School of Mathematics, Watson Building, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom
Address at time of publication: Tessella, 26 The Quadrant, Abingdon Science Park, Abingdon, OX14 3YS, United Kingdom

Received by editor(s): December 19, 2011
Received by editor(s) in revised form: June 26, 2012
Published electronically: February 19, 2014
Additional Notes: The author was supported by an EPSRC doctoral training grant.
This work has appeared previously as part of the author’s MPhil thesis [2]. The author would like to express his gratitude to his MPhil and PhD supervisor, Jonathan Bennett, for all his support and assistance.
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.