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

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.

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 $$\alpha<1/2$$.