超越Young的路径积分：基于Faber–Schauder能量空间
Pathwise integration beyond Young via Faber--Schauder energy spaces
AI总结：
提出基于Faber–Schauder能量空间的路径积分理论，通过二进可和性条件替代经典Young条件，证明积分存在性、有界双线性、可加性、分部积分及Young–Loève估计，并推广到经典Young积分不适用的例子。
摘要：
我们发展了一种基于Faber–Schauder能量空间的路径积分理论。该方法用Faber–Schauder系数表示的二进可和性条件替代了经典的Hölder–Young和有限变差Young条件。在归一化区间$[0,1]$上，这些条件定义了Banach空间$\mathcal{E}^p$，我们称之为Faber–Schauder能量空间。对于满足$1/p+1/q\ge1$的$p,q>1$，我们证明每一对$f\in\mathcal{E}^p$和$g\in\mathcal {E}^q$都允许一个由二进左Riemann和构造的连续路径积分$I_{f,g}$。我们将$I_{f,g}$称为Faber–Schauder积分，并证明它关于$(f,g)$在相应能量范数下有界且双线性。该积分满足可加性、分部积分以及一个二进Young–Loève估计。它也是有限Faber–Schauder逼近的经典Riemann–Stieltjes积分的一致极限。当经典Young积分可用时，Faber–Schauder积分与之一致，但同时也适用于无法验证Hölder–Young条件或有限变差Young条件的确定性和高斯例子。在这个意义上，它提供了基于Faber–Schauder系数的Young框架的推广。
英文摘要：
We develop a pathwise integration theory based on Faber--Schauder energy spaces. The approach replaces the classical Hölder--Young and finite-variation Young conditions by dyadic summability conditions expressed in terms of Faber--Schauder coefficients. On the normalized interval $[0,1]$, these conditions define Banach spaces $\mathcal{E}^p$, which we call Faber--Schauder energy spaces. For $p,q>1$ satisfying $1/p+1/q\ge1$, we prove that every pair $f\in\mathcal{E}^p$ and $g\in\mathcal {E}^q$ admits a continuous pathwise integral $I_{f,g}$, constructed from dyadic left Riemann sums. We call $I_{f,g}$ the Faber--Schauder integral, and show that it depends boundedly and bilinearly on $(f,g)$ in the corresponding energy norms. The integral satisfies additivity, integration by parts, and a dyadic Young--Loève estimate. It is also the uniform limit of classical Riemann--Stieltjes integrals of finite Faber--Schauder approximations. The Faber--Schauder integral agrees with the classical Young integral whenever the latter is available, but also applies to deterministic and Gaussian examples for which neither the Hölder--Young condition nor the finite-variation Young condition can be verified. In this sense, it provides a Faber--Schauder coefficient-based extension of Young's framework.
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