Riesz means in Hardy spaces on Dirichlet groups
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Mathematische Annalen
Riesz means in Hardy spaces on Dirichlet groups Andreas Defant1 · Ingo Schoolmann1 Received: 28 August 2019 / Revised: 17 February 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Given a frequency λ = (λn ), we study when almost all vertical limits of a H1 Dirichlet series an e−λn s are Riesz summable almost everywhere on the imaginary axis. Equivalently, this means to investigate almost everywhere convergence of Fourier series of H1 -functions on so-called λ-Dirichlet groups, and as our main technical tool we need to invent a weak-type (1, ∞) Hardy-Littlewood maximal operator for such groups. Applications are given to H1 -functions on the infinite dimensional torus T∞ , ordinary Dirichlet series an n −s , as well as bounded and holomorphic functions on the open right half plane, which are uniformly almost periodic on every vertical line. Mathematics Subject Classification Primary 43A17; Secondary 30H10 · 30B50
1 Introduction Let λ = (λn ) be a frequency, i.e. a strictly increasing, unbounded sequence of nonnegative real numbers. Moreover, let G be a compact abelian group, and β : R → G a continuous homomorphism of groups with dense range such that for each character e−iλn · : R → T there is a (then unique) character h λn : G → T with e−iλn · = h λn ◦ β. For 1 ≤ p ≤ ∞ denote by H pλ (G) the Hardy space of all f ∈ L p (G) that have a Fourier transform fˆ : Gˆ → T supported by all characters h λn . As recently proven in [6] every f ∈ H pλ (G), 1 < p < ∞, has an almost everywhere convergent Fourier series representation f (ω) = ∞ n=1 f (h λn )h λn (ω). Inspired by the work [11] of Hardy and Riesz on general Dirichlet series from 1915, we in this article study almost everywhere Riesz summability of the Fourier series of
Communicated by Loukas Grafakos.
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Ingo Schoolmann [email protected] Andreas Defant [email protected]
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Institut für Mathematik, Carl von Ossietzky Universität, 26111 Oldenburg, Germany
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A. Defant, I. Schoolmann
functions f ∈ H1λ (G). The main tool is given by an appropriate weak-type (1, ∞) Hardy–Littlewood maximal operator. As a particular case we look at the frequency λ = (log n), the infinite dimensional torus G = T∞ , and the Kronecker flow β : R → G, t → ( pk−it ), where pk denotes the kth prime. Our results prove that each f ∈ H1 (T∞ ) almost everywhere is the pointwise limit of its logarithmic Riesz means, whereas for arithmetic Riesz means (Cesàro means) this in general fails. of our results have equivalent formulations in terms of general Dirichlet series Most an e−λn s an e−λn s . More precisely, almost all vertical limits of Dirichlet series which belong to the Hardy space H1 (λ), are summable by their first Riesz means of any order k > 0 almost everywhere on the imaginary axis [Re = 0] (and consequently at every point on the open right half plane [Re > 0]). λ (G) may be identified with Another application shows, that the Hardy space H∞ the Banach space of all bounded and holomorphic
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