**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][1] at the end of page 9 (they say to see another paper, but I didn't found nothing on it!)

**MY ATTEMPT:** let us take another sequence of partitions whose mash tends to zero, say
$$
\sigma_n=\{s=u_0^n<u_1^n<\cdots<u_{h_n}^n<u_{h_n+1}^n=t\}
$$

and prove that
$$
|M_{ts}^{\pi_n}-M_{ts}^{\sigma_n}|\to0\;\;\;n\to+\infty.
$$

Now
\begin{align*}
M_{ts}^{\pi_n}-M_{ts}^{\sigma_n}
&=\sum_{l=0}^{h_n}B_{u_{l+1}^nr_l^n}
-\sum_{l=0}^{k_n}B_{r_{l+1}^nr_l^n}\\
&=\sum_{l=0}^{k_n}
\left(B_{t_{4l+4}^nt_{4l+2}^n}-B_{t_{4l+3}^nt_{4l+2}^n}-
B_{t_{4l+4}^nt_{4l+3}^n}\right)-
\left(B_{t_{4l+3}^nt_{4l+3}^n}-B_{t_{4l+2}^nt_{4l+1}^n}-
B_{t_{4l+3}^nt_{4l+2}^n}\right)\\
&=\sum_{l=0}^{k_n}
(\delta_2B)_{t_{4l+4}^nt_{4l+3}^nt_{4l+2}^n}-
(\delta_2B)_{t_{4l+3}^nt_{4l+2}^nt_{4l+1}^n}\\
&=\sum_{l=0}^{k_n}
h_{t_{4l+4}^nt_{4l+3}^nt_{4l+2}^n}-
h_{t_{4l+3}^nt_{4l+2}^nt_{4l+1}^n}\\
\end{align*}
supposing wlog that $h_n\le k_n$ and setting, for $l\le h_n$
\begin{align*}
t_{4l+1}^n&:=r_l^n\\
t_{4l+2}^n&:=u_l^n\\
t_{4l+3}^n&:=r_{l+1}^n\\
t_{4l+4}^n&:=u_{l+1}^n\\
\end{align*}
while, for $l>h_n$ the definition of the odd terms stay the same, while $t_{4l+2}^n=t_{4l+4}^n:=t$.

Then we can write
\begin{align*}
|M_{ts}^{\pi_n}-M_{ts}^{\sigma_n}|
&\le\sum_{l=0}^{k_n}
|h_{t_{4l+4}^nt_{4l+3}^nt_{4l+2}^n}|+
|h_{t_{4l+3}^nt_{4l+2}^nt_{4l+1}^n}|\\
&=\sum_{l=0}^{k_n}
\left|\sum_jh_{t_{4l+4}^nt_{4l+3}^nt_{4l+2}^n}^j\right|+
\left|\sum_ih_{t_{4l+3}^nt_{4l+2}^nt_{4l+1}^n}^i\right|\\
&\le\sum_{l=0}^{k_n}\left[
\sum_j\left|h_{t_{4l+4}^nt_{4l+3}^nt_{4l+2}^n}^j\right|+
\sum_i\left|h_{t_{4l+3}^nt_{4l+2}^nt_{4l+1}^n}^i\right|\right]\\
&=\sum_{l=0}^{k_n}\left[
\sum_j\left|\frac{h_{t_{4l+4}^nt_{4l+3}^nt_{4l+2}^n}^j}{|t_{4l+3}^n-t_{4l+2}^n|^{\rho_j}|t_{4l+4}^n-t_{4l+3}^n|^{\mu-\rho_j}}\right|
|t_{4l+3}^n-t_{4l+2}^n|^{\rho_j}|t_{4l+4}^n-t_{4l+3}^n|^{\mu-\rho_j}\right]\\
+&
\left[
\sum_i\left|\frac{h_{t_{4l+3}^nt_{4l+2}^nt_{4l+1}^n}^i}{|t_{4l+2}^n-t_{4l+1}^n|^{\rho_i}|t_{4l+2}^n-t_{4l+2}^n|^{\mu-\rho_i}}\right|
|t_{4l+2}^n-t_{4l+1}^n|^{\rho_i}|t_{4l+3}^n-t_{4l+2}^n|^{\mu-\rho_i}\right]\\
\end{align*}
and since this is true for every $\{h^j\}_j,\{h^i\}_i\subset\mathscr C_3(V)$ such that $\sum_jh^j=\sum_ih^i=h$ and $0<\rho_j,\rho_i<\mu$, passing to the $\inf$ on these parameters, we get 
\begin{align*}
|M_{ts}^{\pi_n}-M_{ts}^{\sigma_n}|
\le
\|h\|_{\mu}\left[\sum_{l=0}^{k_n}\left(\max\{|t_{4l+3}^n-t_{4l+2}^n|,|t_{4l+4}^n-t_{4l+3}^n|\}\right)^{\mu}\\
+\left(\max\{|t_{4l+2}^n-t_{4l+1}^n|,|t_{4l+3}^n-t_{4l+2}^n|\}\right)^{\mu}\right]\\
\le\|h\|_{\mu}\left[\sum_{l=0}^{k_n}\left(|t_{4l+3}^n-t_{4l+2}^n|+|t_{4l+4}^n-t_{4l+3}^n|\right)^{\mu}\\
+\left(|t_{4l+2}^n-t_{4l+1}^n|+|t_{4l+3}^n-t_{4l+2}^n|\right)^{\mu}\right]
\end{align*}

From this, I tried as follows: call $m_0=1$ and take $\pi_{m_0}$; then since the mash tends to zero, there exists $m_1>m_0$ such that $m_0$ elements of $\sigma_{m_1}$, call them $\widetilde{\sigma_{m_1}}:=\{u_{l_s}^{m_1}\}_{s=1}^{m_0}$ are closer as we want to the elements of $\pi_{m_0}$, say $|r_s^{m_0}-u_{j_s}^{m_1}|<\varepsilon_1$.

We have to work on $\sigma_{m_1}$: there exists $m_2>m_1$ such that considering suitable elements of $\pi_{m_2}$, say $\widetilde{\pi_{m_2}}:=\{r_{j_s}^{m_2}\}_{s=1}^{m_1}$, we can approximate the elements of $\sigma_{m_1}$, say $|u_s^{m_i}-r_{j_s}^{m_2}|<\varepsilon_2$, and so on.

Then I did the obvious:
$$
M_{ts}^{\pi_{2\beta}}-M_{ts}^{\sigma_{2\beta+1}}
=\left(M_{ts}^{\pi_{2\beta}}-M_{ts}^{\widetilde{\sigma_{2\beta+1}}}\right)
+\left(M_{ts}^{\widetilde{\sigma_{2\beta+1}}}-M_{ts}^{\widetilde{\pi_{2(\beta+1)}}}\right)
+\left(M_{ts}^{\widetilde{\pi_{2(\beta+1)}}}-M_{ts}^{\sigma_{2(\beta+1)-1}}\right)\;\;;
$$ 
now the first and the last summand go to zero as $\beta\to+\infty$, but the second term doesn't a priori, since it is an approximation of the LHS.


I thought that maybe there is a way to modify the $\widetilde{\pi}$ and $\widetilde{\sigma}$ and/or exploiting the absolute continuity of $M$ in oder to compare the LHS and the second term of RHS, and maybe writing something like 
$$
\left|M_{ts}^{\widetilde{\sigma_{2\beta+1}}}-M_{ts}^{\widetilde{\pi_{2(\beta+1)}}}\right|
=a_{\beta}|M_{ts}^{\pi_{2\beta}}-M_{ts}^{\sigma_{2\beta+1}}|
$$
for some $0<a_{\beta}<1$ such that $\limsup_{\beta\to+\infty}a_{\beta}<1$, thing that allow me to conclude, but for the moment I'm stuck. 




  [1]: https://arxiv.org/pdf/0803.0552.pdf