无穷维多圆盘上$H^\infty(\mathbb{T}^\infty)$中沿递减坐标半径的径向收敛
Radial Convergence along Decreasing Coordinate Radii in \texorpdfstring{$H^\infty(\T^\infty)$}{H-infinity(T-infinity)}
AI总结：
针对无穷维多圆盘上$H^\infty(\mathbb{T}^\infty)$有界解析函数的径向收敛问题，构造反例并证明特定边界点无关逼近的收敛性结论。
AI中文摘要：
Aleman、Olsen和Saksman提出两个问题：无穷维多圆盘上有界解析函数在每个径向点坐标非递增时是否会出现径向收敛失效，且该逼近是否可独立于边界点选取。我们构造了一个依赖于点的反例，其中每个固定坐标仍递增至1；有限离散化得到与边界点无关的逐坐标递减逼近，其向边界函数的收敛几乎处处失效。相比之下，所有与边界点无关且固定坐标单调递增至1的逼近，均满足$1<p<\infty$时的$L^p$极大不等式及几乎处处Fatou定理。
英文摘要：
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 $1<p<\infty$.
