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Fatou and brothers Riesz theorems in the infinite-dimensional polydisc

Aleman, Alexandru LU ; Olsen, Jan Fredrik LU and Saksman, Eero LU (2019) In Journal d'Analyse Mathematique 137(1). p.429-447
Abstract


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
... (More)


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.

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Please use this url to cite or link to this publication:
author
; and
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Journal d'Analyse Mathematique
volume
137
issue
1
pages
429 - 447
publisher
Magnes Press
external identifiers
  • scopus:85062776231
ISSN
0021-7670
DOI
10.1007/s11854-019-0006-x
language
English
LU publication?
yes
id
1d4e6a02-9afd-47b0-953e-1f7f7ab23e4c
date added to LUP
2019-03-19 11:12:08
date last changed
2022-04-25 21:47:44
@article{1d4e6a02-9afd-47b0-953e-1f7f7ab23e4c,
  abstract     = {{<p><br>
                                                         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                             <br>
                            <sup>1</sup><br>
                                                          function almost everywhere to T                             <br>
                            <sup>∞</sup><br>
                                                         . 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.                         <br>
                        </p>}},
  author       = {{Aleman, Alexandru and Olsen, Jan Fredrik and Saksman, Eero}},
  issn         = {{0021-7670}},
  language     = {{eng}},
  number       = {{1}},
  pages        = {{429--447}},
  publisher    = {{Magnes Press}},
  series       = {{Journal d'Analyse Mathematique}},
  title        = {{Fatou and brothers Riesz theorems in the infinite-dimensional polydisc}},
  url          = {{http://dx.doi.org/10.1007/s11854-019-0006-x}},
  doi          = {{10.1007/s11854-019-0006-x}},
  volume       = {{137}},
  year         = {{2019}},
}