变差指数的划分不变性与经典 $p$-变差空间
Partition invariance of variation indices and classical $p$-variation spaces
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- KAIST（韩国科学技术院）
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中文总结 AI 辅助
本文证明在划分的自然几何假设下，沿细化划分序列定义的变差指数与经典变差指数一致，并推广到 Dini 模情形，给出二进 Faber--Schauder 系数刻画，同时通过反例和嵌入结果揭示划分假设的必要性及临界指数处空间的划分依赖性。
中文摘要 AI 辅助
我们研究沿细化划分序列的连续路径的变差指数，该指数定义为使相应的 $p$ 阶变差和一致有界的指数 $p\ge1$ 的下确界。在划分的自然几何假设下，我们证明对于每条具有正 Hölder 正则性的路径，该指数与使用所有有限划分定义的经典变差指数一致。该结果推广到具有任意阶 Dini 模的路径，并给出了经典变差指数在二进 Faber--Schauder 系数方面的刻画。显式反例表明划分假设一般不能省略。然而，即使在这些假设下，在临界指数处相应函数空间的成员资格仍可能依赖于划分序列。我们进一步获得进入具有消失经典 $p$-变差的路径空间的紧致次临界嵌入，并建立了系数空间与变差空间的严格层级关系。最后，我们证明经典 $p$-变差空间到具有一致有界二进 $p$ 阶变差和的连续路径空间的连续嵌入具有非闭值域。
英文摘要
We study the variation index of a continuous path along a refining partition sequence, defined as the infimum of exponents $p\ge1$ for which the corresponding $p$-th variation sums are uniformly bounded. Under natural geometric assumptions on the partitions, we prove that this index coincides with the classical variation index, defined using all finite partitions, for every path with positive Hölder regularity. The result extends to paths with an all-orders Dini modulus and yields a characterization of the classical variation index in terms of dyadic Faber--Schauder coefficients. Explicit counterexamples show that the partition assumptions cannot in general be omitted. Even under these assumptions, however, membership in the corresponding function spaces at the critical exponent may still depend on the partition sequence. We further obtain compact subcritical embeddings into the space of paths with vanishing classical $p$-variation and establish a strict hierarchy of coefficient and variation spaces. Finally, we show that the continuous embedding of the classical $p$-variation space into the space of continuous paths with uniformly bounded dyadic $p$-th variation sums has nonclosed range.
