Variants of a theorem of Helson on general Dirichlet series

Abstract

A result of Helson on general Dirichlet series a n e −λns states that, whenever (a n ) is 2-summable and λ = (λ n ) satisfies a certain condition introduced by Bohr, then for almost all homomorphism ω : (R, +) → T the Dirichlet series a n ω(λ n )e −λns converges on the open right half plane [Re > 0]. For ordinary Dirichlet series a n n −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 H p of such series. The aim here is to prove variants of Helson's theorem within our recent theory of Hardy spaces H p (λ), 1 ≤ p ≤ ∞, of general Dirichlet series. To be more precise, in the reflexive case 1 < p < ∞ we extend Helson's result to Dirichlet series in H p (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 give several applications to the structure theory of Hardy spaces of general Dirichlet series.

Henry Helson meets other big shots — a brief survey

“…Again we may deduce convergence theorems. Combining Theorem 5.2 and adding Lemma 2.1, we conclude the following result from [11] (see again [19,Theorem 1.4] for the case p = 2). This is a considerably strong extension of Theorem 1.1.…”

“…In particular this holds for D itself. By (11) we know that the functions f : R → C, f χ (t) = f (χ(p)p −it ) are integrable for almost all χ ∈ Ξ. Comparing Fourier coefficients we conclude that…”

“…Clearly, here the particular case p = 2 proves Theorem 3.1 for u = 0 which means a strong improvement. In fact the proof from [14] (and also [11]) follows the strategy invented by Hedenmalm-Saksman in [19]; this strategy applies the orginal Carleson-Hunt theorem for functions in one variable through a magic trick invented by Fefferman in [15]. Observe, that if the function f in the preceding theorem only depends on the first variable, then we recover the full one variable case from Theorem 5.1.…”

“…Can Helson-like arguments be used to prove the important Banach space identity H ∞ = D ∞ from (9)? Both questions have a positive answer, and this will be a consequence of the following maximal inequality from [11] which is a variant of Theorem 5.2. Theorem 6.1.…”

Hardy spaces of general Dirichlet series — a survey

“…The following theorem from [25] is a far reaching extension of Helson's theorem. It in fact shows that, given an appropriate frequency λ, Helson's theorem extends to functions f from our scale H λ p (G), 1 ≤ p < ∞, modelled on λ-Dirichlet groups G. Moreover, it describes the G-a.e.…”