实现平方函数度量中的控制测度界
Majorizing-measure bounds in the realized square-function metric
浏览论文内容
中文总结 AI 辅助
本文在实现平方函数度量下直接进行控制测度论证，为光滑巴拿赫空间中的可预测高斯和建立了极大不等式，结合局部指数不等式、路径平均与停时，并推广至熵与连续性估计。
中文摘要 AI 辅助
可预测平方函数衡量鞅的大小。应用于参数差异时，它同样定义了一个随机伪度量。我们解释了如何直接在此实现几何中进行控制测度论证，而无需将终端域条件化为高斯分布，也无需先将度量替换为确定性度量的随机倍数。对于在$(2,D)$-光滑巴拿赫空间中的有限族可预测高斯和，以及参数集上的固定概率测度$\u0000$，我们证明了对于$p\ue1e1$，有$\norm{\u0000max_j\ue000osc_T f_j}_{L^p} \ue000le CD\norm{\ue000G_\u0000(d)+\ue000sqrt p\u0000,\u0000Delta_d}_{L^p}$，其中$\ue000Delta_d$是参数空间在终端平方函数度量下的直径，$\ue000G_\u0000(d)$是其球质量积分。证明结合了局部化指数不等式、随机球上的路径平均以及停时论证。原子测度恢复了对数加权的极大不等式；哈尔测度将结果与齐次熵和连续性估计联系起来。我们还给出了推广、示例以及关于选择平均测度限制的精确说明。
英文摘要
The predictable square function measures the size of a martingale. Applied to parameter differences, it also defines a random pseudometric. We explain how to perform a majorizing-measure argument directly in this realized geometry, without conditioning the terminal field to be Gaussian and without first replacing the metric by a random multiple of a deterministic one. For a finite family of predictable Gaussian sums in a $(2,D)$-smooth Banach space, and a fixed probability measure $μ$ on the parameter set, we prove \[ \norm{\max_j\osc_T f_j}_{L^p} \le CD\norm{\G_μ(d)+\sqrt p\,Δ_d}_{L^p},\qquad p\ge1, \] where $Δ_d$ is the diameter of the parameter space in the terminal square-function metric and $\G_μ(d)$ is its ball-mass integral. The proof combines a localized exponential inequality, pathwise averaging over random balls, and a stopping-time argument. Atomic measures recover logarithmically weighted maximal inequalities; Haar measure connects the result with homogeneous entropy and continuity estimates. We also give extensions, examples, and a precise account of the restrictions on choosing the averaging measure.
