THE FRAMEWORK: let us consider a real topological vector space $V$.

We denote with $\mathscr C_k(V)$ the set of all continous functions $f:[0,T]^k\to V$ such that $f_{t_1\cdots t_k}=0$ whenever $t_i=t_{i+1}$ for some $0\le i\le k-1$.

We define the operator $\delta_k:\mathscr C_k(V)\to\mathscr C_{k+1}(V)$ as follows: $$ (\delta_kf)_{t_1\cdots t_{k+1}}:=\sum_{j=1}^{k+1}(-1)^jf_{t_1\cdots \widehat t_{j}\cdots t_{k+1}} $$ where the hat $\widehat{\cdot}$ means that argument is omitted.

So if $f\in\mathscr C_1(V)$ then $(\delta_1f)_{ts}=f_t-f_s$ and if $g\in\mathscr C_2(V)$ then $(\delta_2g)_{tus}=-g_{us}+g_{ts}-g_{tu}$.

Next we define the following norms: if $g\in\mathscr C_2(V)$ then, for $\mu>1$ we set $$ \|g\|_{\mu}:=\sup_{0\le s<t\le T}\frac{|g_{st}|}{|t-s|^{\mu}} $$ while for $h\in\mathscr C_3(V)$ we first set $$ \|h\|_{\mu,\rho}:=\sup_{0\le s<t\le T}\frac{|h_{tus}|}{|t-u|^{\rho}|u-s|^{\mu-\rho}} $$ (here $0<\rho<\mu$) and then define the norm: $$ \|h\|_{\mu}:=\inf\left\{\sum_j\|h_j\|_{\rho_j,\mu-\rho_j}:h=\sum_jh_j,\; h_j\in\mathscr C_3,0<\rho_j<\mu\right\}\;. $$

Finally let us denote for $k=2,3$ $$ \mathscr C_k^{\mu}:=\{h\in\mathscr C_k:\|h\|_{\mu}<+\infty\}. $$ and accept that $$ \ker\delta_k=\operatorname{Im}\delta_{k-1} $$ (this holds for every $k$).

THE PROBLEM: let us take $h\in\mathscr C_3^{\mu}$ such that $\delta_3h=0$. Then it is not difficult to prove that there exists $B\in\mathscr C_2$ such that $\delta_2B=h$. Now fix $0\le s<t\le T$ and consider on $[s,t]$ a sequence of partitions $\{\pi_n\}_n$ whose mash tends to zero.

To fix ideas we write $$ \pi_n=\{s=r_0^n<r_1^n<\cdots<r_{k_n}^n<r_{k_n+1}^n=t\} $$

Define then $$ M_{ts}^{\pi_n}:=B_{ts}-\sum_{l=0}^{k_n}B_{r_{l+1}^nr_l^n}\;. $$ Now accept this last one converges (up to passing to a subsequence): how can we show that the limit does not depend on the particular sequence of partitions chosen?

In the next minutes I'll write my attempt, however in the meanwhile I post my problem, so if someone has some good hint, I will thank you!

PS: this is taken from a proof contained in the 2010 paper by M.Gubinelli and S. Tindel ROUGH EVOLUTION EQUATIONS at the end of page 9 (they say to see another paper, but I didn't found nothing on it!)