Let $N \geq \lfloor 1/\alpha \rfloor > 0$ and consider a weakly geometric $\alpha$-Hölder rough path $\textbf{x}$ that preserves the origin, i.e. an element $\textbf{x} \in C^{\alpha\text{-Höl}}_o([0,T],G^{\lfloor 1/\alpha \rfloor}(\mathbb{R}^d))$. This is standard notation, and is the one in the book "Multidimensional Stochastic Processes as Rough Paths: Theory and Applications" by Peter Friz and Nicolas Victoir.
It is well know that there exists a unique Lyons lift of order $N$, i.e for each $\textbf{x} \in C^{\alpha\text{-Höl}}_o([0,T],G^{\lfloor 1/\alpha \rfloor}(\mathbb{R}^d))$ there exists a unique $$S_N(\textbf{x}) \in C^{\alpha\text{-Höl}}_o([0,T],G^{N}(\mathbb{R}^d)).$$ See Theorem 9.5 in the book above for a more precise statement. My question is about the estimates that one gets. In particular, we know that there exists a constant $C(N,\alpha)$ such that $$||S_N(\textbf{x})||_{\alpha\text{-Höl};[s,t]} \leq C(N,\alpha) ||\textbf{x}||_{\alpha\text{-Höl};[s,t]}.$$ The estimate above holds for every $N$. But what about the whole signature? Specifically, let $S_\infty(\textbf{x})$ denote the entire signature associated to $\textbf{x}$. Can we bound the $\alpha$-Hölder norm of $S_\infty(\textbf{x})$ with $||\textbf{x}||_{\alpha\text{-Höl};[s,t]}$? In other words, can we obtain the estimate above with $S_N$ substituted by $S_\infty$?
Yes, we see this by following their definition of $C$ in the proofs.
In lemma 9.2, they bound in (9.1) by $\frac{2^{N}}{(N+1)!}\ell^{N+1}=B_{N}\ell^{N+1}$. This $B_{N}\to 0$ as $N\to +\infty$. Then in the proof of 9.3, they use this same constant $B_{N}$ a couple of times due to the four triangle inequality arguments. So at worst we get some $(cB_{N})^{a}$ for constants $c,a>0$. They also use there the constant $C_{N,p}=1/(1-2^{1-(N+1)/p})\to 1$. So overall all the constants are bounded as $N\to +\infty$.