Fatou and brothers Riesz theorems in the infinite-dimensional polydisc (Q2000398)

The infinite-dimensional torus \(\mathbb{T}^{\infty}\) is the countable product of \(\mathbb{T} = \{w \in \mathbb{C}\colon |w|=1\}\), and the Haar measure on this compact abelian group is the normalized Lebesgue measure. The Hardy space \(H_{1}(\mathbb{T}^{\infty})\) is defined as the closed subspace of all \(f \in L_1(\mathbb{T}^\infty)\) such that the Fourier coefficient \( \widehat{f}(\alpha) = \int_{\mathbb{T}^{\infty}} f(w) w^{-\alpha} dw =0\,, \) whenever \(\alpha\) is a finite sequence of integers with at least one negative entry. It is well known that, by convolution with the Poisson kernel, every \(f \in H_1(\mathbb{T})\) extends to some \(F\) in the Hardy space \( H_1(\mathbb{D})\) of all holomorphic functions on the disc \(\mathbb{D}\). Moreover, in a sense conversely, by Fatou's theorem, every \(F \in H_1(\mathbb{D})\) has radial (and even non-tangential) limits almost everywhere on \(\mathbb{T}\), i.e., for almost all \(z \in \mathbb{T}\) we have \(F(rz) \to f(z)\) as \(r \uparrow 1\). To extend all this to the infinite-dimensional torus \(\mathbb{T}^{\infty}\) is highly problematic, and the purpose of this interesting article is to initiate the investigation to which extent such a Fatou-type approximation holds in the infinite-dimensional setting. To see a sample, recall that every \(f \in H_1(\mathbb{T}^{\infty})\) extends to a holomorphic function \(F\) on the intersection \(\ell_2 \cap B_{c_0}\) such that each Fourier coefficient \(\widehat{f}(\alpha)\) coincides with the monomial coefficient \(\partial^\alpha F(0)/\alpha!\). Then it is proved here that, for almost all \(z \in \mathbb{T}^\infty\), we have that \(F( (r^k z_k)_{k=1}^\infty) \to f(z)\) as \(r \uparrow 1\). Replacing functions \(f \in H_1(\mathbb{T}^{\infty})\) by Borel measures on \(\mathbb{T}^\infty\) with Fourier spectrum in \(\mathbb{N}_0^\infty \cup (-\mathbb{N}_0^\infty)\), all this is studied in a much more general framework. As a consequence, the authors obtain a new proof of the brothers Riesz theorem for analytic measures on \(\mathbb{T}^{\infty}\) (originally due to Helson and Lowdenslager). Moreover, a counterexample is provided showing that Fatou's theorem is not true in infinite dimensions without restrictions to the mode of radial convergence. The article finishes with an interesting list of open problems. Finally, we remark that, due to an ingenious observation of H. Bohr, harmonic analysis on \(\mathbb{T}^{\infty}\) is intimately related with the theory of ordinary Dirichlet series \(\sum a_n n^{-s}\). This led in recent years to a substantial revival of interest in ordinary Dirichlet series (see, e.g., the recent monographs [\textit{A. Defant} et al., Dirichlet series and holomorphic functions in high dimensions. Cambridge: Cambridge University Press (2019; Zbl 1460.30004)] and [\textit{H. Queffélec} and \textit{M. Queffélec}, Diophantine approximation and Dirichlet series. New Delhi: Hindustan Book Agency (2013; Zbl 1317.11001)]).