Let $X^n$ be a collection of smooth functions so that their $\alpha$-Holder norms are uniformly bounded

Question

Let $X^n$ be a collection of smooth functions so that their $\alpha$-Holder norms for $\alpha \in (1/3,1/2)$ are uniformly bounded - that is $\sup_n \|X^n\|_\alpha<\infty$. Define the standard Riemann-Stieltjes integrals $\mathbb X_{s,t}^n=\int_s^t (X^n(r)-X^n(s) )\otimes dX^n(r)$. Then is it true that $\sup_n\|\mathbb X^n\|_{2\alpha}<\infty$?

This is related to rough paths theory and the interpolation result of Friz and Hairer's exercise 2.9. I am wondering if the second condition really needs to be checked.

Answer 1

This cannot hold, and in a sense rough path theory has to be developed precisely because of this reason; otherwise, rough path lifts would be defined uniquely for any curve of Hölder regularity $>1/3$.

For a simple example, take $Y_n:t\mapsto n^{-1}\exp(in^2t)$, which converges to zero in every $(1/2-\varepsilon)$-Hölder norm (for instance by looking at times $|t-s|\geq1/n^2$ or $|t-s|\leq1/n^2$). The component of $\mathbb Y_{0,1}$ in $1\otimes i$ converges to $1/2$, so $X_n=Y_n/\|Y_n\|_\alpha$ is a counterexample for all $\alpha<1/2$.