解决 Gilbert 猜想：取值于 Clifford 模的 Hardy 空间中的维数依赖
Resolving Gilbert's Conjecture: Dimensional Dependencies in Hardy Spaces Valued in Clifford Modules
AI总结：
本文完整解决了 1991 年 Gilbert 猜想，证明取值于 Clifford 模的高维 Hardy 空间表示仅在 $n\not\equiv6,7\pmod8$ 时成立，并借助八元数与 Riesz 变换给出代数刻画和显式实现。
摘要：
本文对 Gilbert 猜想进行了全面研究，该猜想涉及上半空间中取值于 Clifford 模的 Hardy 空间。我们探讨了 Gilbert 于 1991 年提出的猜想，其目标是把经典原理推广到取值于任意 Clifford 模的高维 Euclid 空间；该经典原理将实直线上的实值 $L^p$ 函数表示为 Hardy 全纯函数的边值。我们给出了这一猜想的完整解答，证明其有效性取决于维数 $n$：具体而言，当 $n\not\equiv 6,7\pmod 8$ 时猜想成立，在其他情况下则不成立。本文强调了一个关键发现，即 Gilbert 猜想可以重新表述为一组代数条件。为处理这些条件，我们采用了一种利用 octonions（八元数）的新颖策略，揭示了它们在解决与 Clifford 模和旋量相关问题时的工具性作用。这一创新方法不仅通过把 Hilbert 变换推广为 Riesz 变换给出了显式实现，也在理解高维 Hardy 空间方面建立了一项重要进展。
英文摘要：
This article provides a thorough investigation into Gilbert's Conjecture, pertaining to Hardy spaces in the upper half-space valued in Clifford modules. We explore the conjecture proposed by Gilbert in 1991, which seeks to extend the classical principle of representing real $L^p$ functions on the real line as boundary values of Hardy holomorphic functions to higher-dimensional Euclidean spaces valued in any Clifford module. We present a complete resolution to this conjecture, demonstrating that its validity is contingent upon the dimension $n$, specifically holding true when \(n \not\equiv 6, 7 \mod 8\) and failing otherwise. The pivotal discovery that Gilbert's conjecture can be reformulated as a set of algebraic conditions is underscored in this work. To navigate these conditions, we employ a novel strategy that leverages the octonions, revealing their instrumental role in addressing issues related to Clifford modules and spinors. This innovative approach not only provides explicit realization through the generalization of the Hilbert transform to the Riesz transform but also establishes a significant advancement in the understanding of Hardy spaces within higher dimensions.
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