Fatou and brothers Riesz theorems in the infinite-dimensional polydisc

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We study the boundary behavior of functions in the Hardy spaces on the infinite-dimensional polydisc. These spaces are intimately related to the Hardy spaces of Dirichlet series. We exhibit several Fatou and Marcinkiewicz- Zygmund type theorems for radial convergence of functions with Fourier spectrum supported on N0∞∪(−N0∞). As a consequence one obtains easy new proofs of the brothers F. and M. Riesz Theorems in infinite dimensions, as well as being able to extend a result of Rudin concerning which functions are equal to the modulus of an H 1 function almost everywhere to T . 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.

Sidor (från-till)429-447
TidskriftJournal d'Analyse Mathematique
Tidigt onlinedatum2019
StatusPublished - 2019

Ämnesklassifikation (UKÄ)

  • Matematisk analys


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