Variants of a theorem of Helson on general Dirichlet series

Abstract

A result of Helson on λ-Dirichlet series ∑ane−λns states that, whenever (an) is 2-summable and λ=(λn) satisfies a certain condition introduced by Bohr, then for almost all homomorphism ω:(R,+)→T the Dirichlet series ∑anω(λn)e−λns converges on the open right half plane [Re>0]. For ordinary Dirichlet series ∑ann−s Hedenmalm and Saksman related this result with the famous Carleson-Hunt theorem on pointwise convergence of Fourier series, and Bayart extended it within his theory of Hardy spaces Hp of such series. The aim here is to prove variants of Helson's theorem within our recent theory of Hardy spaces Hp(λ),1≤p≤∞, of λ-Dirichlet series. To be more precise, in the reflexive case 1<p<∞ we extend Helson's result to Dirichlet series in Hp(λ) without any further condition on the frequency λ, and in the non-reflexive case p=1 to the wider class of frequencies satisfying the so-called Landau condition (more general than Bohr's condition). In both cases we add relevant maximal inequalities. Finally, we try to indicate how our study of Helson's theorem can be employed to understand that for a fixed but arbitrary frequency λ certain key stones of the structure theory of λ-Dirichlet series in fact turn out to be equivalent.