Fatou and brothers Riesz theorems in the infinite-dimensional polydisc

Research output: Contribution to journalArticle


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.


External organisations
  • University of Helsinki
Research areas and keywords

Subject classification (UKÄ) – MANDATORY

  • Mathematical Analysis
Original languageEnglish
Pages (from-to)429-447
JournalJournal d'Analyse Mathematique
Issue number1
Early online date2019
Publication statusPublished - 2019
Publication categoryResearch