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Radial Convergence along Decreasing Coordinate Radii in \texorpdfstring{H∞(T∞)}{H-infinity(T-infinity)}
Abstract
Aleman, Olsen, and Saksman asked whether radial convergence may fail for bounded analytic functions on the infinite-dimensional polydisc when every radial point has non-increasing coordinates, and whether the approach may be chosen independently of the boundary point. We construct a point-dependent counterexample in which every fixed coordinate nevertheless increases to $1$. A finite discretization yields a boundary-point-independent coordinatewise decreasing approach for which convergence to the boundary function fails almost everywhere. In contrast, every boundary-point-independent approach whose fixed coordinates increase monotonically to $1$ satisfies an Lp maximal inequality and the almost-everywhere Fatou theorem for 1<p<∞.
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BibTeX
@misc{sun2026radial,
title = {{Radial Convergence along Decreasing Coordinate Radii in \texorpdfstring$H^\infty(\T^\infty)$H-infinity(T-infinity)}},
author = {Jiawei Sun and Chao Zu and Yufeng Lu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.03164},
eprint = {2608.03164},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.03164},
}
RIS
TY - UNPB
TI - Radial Convergence along Decreasing Coordinate Radii in \texorpdfstring{H^∞(T^∞)}{H-infinity(T-infinity)}
AU - Jiawei Sun
AU - Chao Zu
AU - Yufeng Lu
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2608.03164
UR - https://arxiv.org/abs/2608.03164
AB - Aleman, Olsen, and Saksman asked whether radial convergence may fail for bounded analytic functions on the infinite-dimensional polydisc when every radial point has non-increasing coordinates, and whether the approach may be chosen independently of the boundary point. We construct a point-dependent counterexample in which every fixed coordinate nevertheless increases to $1$. A finite discretization yields a boundary-point-independent coordinatewise decreasing approach for which convergence to the boundary function fails almost everywhere. In contrast, every boundary-point-independent approach whose fixed coordinates increase monotonically to $1$ satisfies an L^p maximal inequality and the almost-everywhere Fatou theorem for 1<p<∞.
ER -