Vol. 292, No. 1, 2018

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ISSN: 0030-8730
Pointwise convergence of almost periodic Fourier series and associated series of dilates

Christophe Cuny and Michel Weber

Vol. 292 (2018), No. 1, 81–101
Abstract

Let S2 be the Stepanov space with norm fS2 = supx(xx+1|f(t)|2dt)12, λn , and let (an)n1 satisfy Wiener’s condition [b] n1( k:nλkn+1|ak|)2 < . We establish the following maximal inequality:

supN1| n=1Na neiλnt| S2 C( n1( k:nλkn+1|ak|)2)12,

where C > 0 is a universal constant. Moreover, the series n1aneitλn converges for λ-a.e. t . We give a simple and direct proof. This contains as a special case Hedenmalm and Saksman’s result for Dirichlet series. We also obtain maximal inequalities for corresponding series of dilates. Let (λn)n1, (μn)n1, be nondecreasing sequences of real numbers greater than 1. We prove the following interpolation theorem. Let 1 p,q 2 be such that 1p + 1q = 3 2. There exists C > 0 such that for any sequences (αn)n1 and (βn)n1 of complex numbers such that n1( k:nλk<n+1 |αk|)p < and n1( k:nμk<n+1 |βk|)q < , we have

supN1| n=1Nα n D(λnt)|S2 C( n1 ( k:nλk<n+1 |αk |)p)1p ( n1 ( k:nμk<n+1 |βk |)q)1q ,

where D(t) = n1βneiμnt is defined in S2. Moreover, n1αnD(λnt) converges in S2 and for λ-a.e. t . We further show that if {λk,k 1} satisfies the condition

k,k (k,)(k,) (1 |(λk λ) (λk λ)|)+2 < ,

then the series kakeiλkt converges on a set of positive Lebesgue measure only if the series k=1|ak|2 converges. The above condition is in particular fulfilled when {λk,k 1} is a Sidon sequence.

Keywords
almost periodic function, Stepanov space, Carleson theorem, Dirichlet series, dilated function, series, almost everywhere convergence
Mathematical Subject Classification 2010
Primary: 42A75
Secondary: 42A24, 42B25
Milestones
Received: 11 August 2016
Accepted: 31 March 2017
Published: 22 September 2017
Authors
Christophe Cuny
Université de la Nouvelle-Calédonie
Nouméa
New Caledonia
Michel Weber
IRMA, UMR 7501
Strasbourg
France