Pathwise integration beyond Young via Faber--Schauder energy spaces

Abstract

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.