[Paper Review] Fatou and brother Riesz theorems in the infinite-dimensional polydisc

Alexandru Aleman, Jan‐Fredrik Olsen+1 more|arXiv (Cornell University)|Dec 4, 2015

Stochastic processes and financial applications12 references3 citations

TL;DR

This paper establishes infinite-dimensional analogues of Fatou and brothers Riesz theorems for Hardy spaces on the infinite-dimensional polydisc, proving radial boundary limits exist for functions with Fourier spectra supported on non-negative and non-positive multi-indices. It provides new proofs of the brothers Riesz theorems and extends Rudin's result on moduli of $ H^1 $ functions, while constructing counterexamples showing pointwise radial limits fail without spectral restrictions.

ABSTRACT

We study the boundary behavior of functions in the Hardy spaces on the infinite dimensional polydisk. These spaces are intimately related to the Hardy spaces of Dirichlet series. We exhibit several Fatou and Marcinkiewicz-Zygmund type theorems for radial convergence. As a consequence one obtains easy new proofs of the brothers F. and M. Riesz theorems in infinite dimension. Finally, we provide counterexamples showing that the pointwise Fatou theorem is not true in infinite dimensions without restrictions to the mode of radial convergence even for bounded analytic functions.

Motivation & Objective

• To extend classical Fatou and Marcinkiewicz-Zygmund theorems to the infinite-dimensional polydisc setting.
• To provide new, simplified proofs of the infinite-dimensional brothers F. and M. Riesz theorems using radial convergence results.
• To generalize Rudin's result on $ H^1 $ functions being equal a.e. to the modulus of another $ H^1 $ function to $ bT^$.
• To investigate the failure of pointwise radial boundary limits in infinite dimensions and construct counterexamples.
• To identify conditions under which radial boundary behavior remains valid despite the lack of finite-dimensional regularity.

Proposed method

• Analyzes boundary behavior of $ H^p(bT^) $ functions via radial approach along the sequence $ (r^n e^{i heta_n}) $ with $ r \nearrow 1 $, focusing on functions with Fourier spectra in $ bN^_0 \cup (-\bbN^_0) $.
• Applies weak-* convergence and maximal function estimates for Poisson and conjugate Poisson kernels on each torus factor $ bT $, using $ Mu_n \geq C \min(1, \delta_n |t|^{-1}) $ to derive divergence of series.
• Employs the Herglotz representation and $ \ell^2 $-based estimates to show uniform boundedness and non-vanishing of $ f $ on $ c_0 \cap \bbD^\f $, ensuring analyticity in the polydisc.
• Uses independence and uniform boundedness of $ u_n $ and $ \tilde{u}_n $, along with [7, Lemma 3.14], to deduce a.e. divergence of $ \sum Mu_n $ and $ \sum |\tilde{u}_n| $, implying divergent radial paths.
• Constructs a bounded analytic function $ f \in H^\infty(\bbT^\f) $ such that radial limits fail a.e. along a specific radial approach, using perturbed radii $ r_{n,k} $ depending on index blocks.
• Applies Hilbert's argument to show $ f $ has bounded point evaluations on $ c_0 \cap \bbD^\f $, ensuring $ f $ is well-defined and non-vanishing in the polydisc.

Experimental results

Research questions

• RQ1 Can the spectral restriction in Theorem 1 (Fourier support on $ \bbN^\f_0 \cup (-\bbN^\f_0) $) be removed?
• RQ2 Does there exist a bounded analytic function on $ \bbT^\f $ for which radial convergence fails even under monotonic radii $ r_{n,k} \geq r_{n+1,k} $?
• RQ3 Is radial convergence failure unavoidable under pointwise-independent radial approaches?
• RQ4 What is the behavior of non-tangential boundary limits for $ H^p(\bbT^\f) $ functions?
• RQ5 For which radial approaches does the brothers Riesz theorem (vanishing boundary values a.e. implies $ f \equiv 0 $) hold?

Key findings

• Theorem 1 establishes radial boundary limits for $ f \in H^p(\bbT^\f) $ with Fourier spectrum in $ \bbN^\f_0 \cup (-\bbN^\f_0) $, extending classical Fatou-type results to infinite dimensions.
• Corollary 1 and Corollary 2 provide new, simplified proofs of the infinite-dimensional brothers F. and M. Riesz theorems using radial convergence techniques.
• Corollary 3 characterizes $ f \in H^1(\bbT^\f) $ via uniform $ L^1 $-boundedness of its $ m $-th partial products, offering a new criterion for integrability.
• Theorem 2 generalizes the Marcinkiewicz-Zygmund theorem to infinite dimensions, showing that singular measures with such Fourier support have vanishing radial limits a.e.
• Theorem 4 constructs a bounded analytic function $ f \in H^\infty(\bbT^\f) $ such that radial limits fail almost everywhere under a specific radial approach, demonstrating the necessity of spectral restrictions.
• The counterexample relies on a divergent series $ \sum Mu_n = \infty $ a.e., constructed via $ \delta_n \sim 1/(n+2)\log(n+2) $, leading to unbounded radial growth in $ \arg f $.