Representing Measures and Hardy Spaces for the Infinite Polydisk Algebra

Abstract

Representing measures and Jensen measures are studied for the uniform algebra on the infinite polydisk Δ¯∞ generated by the coordinate functions z1 z2,…. Let σ be the Haar measure on the infinite torus T∞, which is the distinguished boundary of the infinite polydisk. For fixed p in the range 1 < p, < ∞, it is shown that a point ζ ∈ Δ∞ has a representing measure in LP(σ) if and only if ζ ∈ l2. A related result for a class of representing measures for the origin, including the Haar measure σ, and for fixed p in the range 0 < p < ∞, is that the point evaluation at ζ is continuous in the Lp‐norm if and only if ζ ∈ Δ∞ ∩l2. In this case the functions in Hp are shown to correspond to analytic functions on the domain Δ∞ l2 in l2. Along these same lines, it is shown that H∞ (σ) is isometrically isomorphic to H (Δ∞ ∩ l2), and also to H∞(B), where B is the open unit ball of the sequence space c0.

The Bohr inequality for ordinary Dirichlet series

“…The determination of the isometric Bohr abscissa ̺ 1 (E) is obviously more delicate, and whereas we are able to determine ̺(E) for the preceding spaces, we will content ourselves, apart from two exceptions, with estimates for ̺ 1 (E) (observe that ̺ 1 (E) ≥ ̺(E)). One of our theorems shows that ̺ 1 (H ∞ ) < ∞, whereas there is no Bohr phenomenon for power series in infinitely many variables and Bohr's theory shows that H ∞ can be seen as a space of Taylor series on the infinite polydisc (see also [15]), so one might think that there is no such phenomenon for Dirichlet series. Comparing with [18], we shall also examine the effect of replacing the space ℓ 1 by ℓ p , 1 ≤ p ≤ 2, and comparing with (3) and (4) ( [24]), we shall examine the effect of taking a 1 = 0, or of replacing |a 1 | by |a 1 | 2 , without affecting the other terms.…”

Weak Product Spaces of Dirichlet Series

“…In view of (17) and Lemma 7 we obtain (18) for f with constant term a 1 = 0, that is, for f ∈ H p 0 . Note that the linear functional f → a 1 is bounded on H p , corresponding to the functional Bf → Bf (0) on H p (T ∞ ) [12]. Hence, the closed subspace H p 0 is complemented in H p by C, and (18) follows in general for f ∈ H p , with one side being finite if and only if the other is.…”

Paley-Wiener Isomorphism Over Infinite-Dimensional Unitary Groups

“…Needed properties of Hardy spaces H 2 χ can be found in [15]. Various cases of Hardy spaces in infinite-dimensional settings were considered in [9,17]. Now, we briefly describe results.…”