Fatou and brother Riesz theorems in the infinite-dimensional polydisc
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.
Keywords
Cite
@article{arxiv.1512.01509,
title = {Fatou and brother Riesz theorems in the infinite-dimensional polydisc},
author = {Alexandru Aleman and Jan-Fredrik Olsen and Eero Saksman},
journal= {arXiv preprint arXiv:1512.01509},
year = {2017}
}
Comments
17 pages. In this version, Theorem 1 has been corrected. As a consequence, Theorem 2 and Corollary 4 have been modified, and Corollary 5 (in the enumeration of the previous version) has been removed