Stieltjes时钟下集值映射的选择：正则性、变差与原子结构
Selections of Set-Valued Maps under Stieltjes Clocks: Regularity, Variation, and Atomic Structure
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中文总结 AI 辅助
本文研究Stieltjes时钟下紧值集值映射的单值选择，证明在g-Hölder指数≥1时可保留正则性与变差，刻画跳跃结构及零维时钟像的充分必要条件。
中文摘要 AI 辅助
Stieltjes时钟允许有效时间连续推进、在区间上保持不变或发生跳跃。我们研究相对于Stieltjes时钟演化的紧值集值映射是否允许一个单值选择，该选择通过一个指定的图形点，同时保留多函数的正则性和变差。对于g-Hölder指数$\alpha \geq 1$，我们证明可以在不增加g-Hölder半范数的情况下保持正则性，且无需对g作单调性假设。若g额外非递减，则一个指定点选择在每个子区间上同时保留该正则性和Hausdorff变差。因此，同一选择保留了与非递减外部时钟相关的所有有限Riesz p-变差。对于$\alpha>1$，结构变为跳跃驱动。对于左连续非递减Stieltjes时钟，连续时钟演化不能产生变差：所有变差由跳跃承载。我们获得了集值映射及其选择的精确跳跃分解，以及Riesz p-变差的显式原子公式。例子表明主要正则性、变差和跳跃界均可达到。对于紧凸欧几里得值映射，我们还考察了$0<\alpha<1$的情形。在此设置中，Hölder指数仍可保留，但通过指定点保留相同常数在一维中成立，而在高维中可能失败。最后，在没有定量Hölder界的情况下，我们精确刻画了何时每个紧值Hausdorff g-连续映射都允许一个指定点g-连续选择：恰好当时钟像为零维。
英文摘要
A Stieltjes clock allows effective time to advance continuously, remain unchanged over intervals, or jump. We study whether a compact-valued set-valued map evolving relative to a Stieltjes clock admits a single-valued selection that passes through a prescribed graph point while retaining the regularity and variation of the multifunction. For g-Hölder exponents $α\geq 1$, we prove that regularity can be preserved without increasing the g-Hölder seminorm, with no monotonicity assumption on g. If g is additionally nondecreasing, one prescribed-point selection preserves both this regularity and the Hausdorff variation on every subinterval. The same selection consequently preserves all finite Riesz p-variations associated with nondecreasing external clocks. For $α>1$, the structure becomes jump-driven. For left-continuous nondecreasing Stieltjes clocks, continuous clock evolution cannot generate variation: all variation is carried by jumps. We obtain exact jump decompositions for both the set-valued map and its selection, together with explicit atomic formulas for Riesz p-variation. Examples show that the principal regularity, variation, and jump bounds are attained. For compact-convex Euclidean-valued maps, we also examine the case $0<α<1$. In this setting, the Hölder exponent can still be preserved, but preservation of the same constant through a prescribed point holds in one dimension and can fail in higher dimensions. Finally, without a quantitative Hölder bound, we characterize exactly when every compact-valued Hausdorff g-continuous map admits a prescribed-point g-continuous selection: precisely when the clock image is zero-dimensional.
发表机构
- Istanbul University（伊斯坦布尔大学）
- Istinye University（伊斯蒂耶大学）
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