mean it is a continuous, non-negative, function that is super-additive and satisfies $\omega(t,t) = 0$ for all $t \in [0,T]$. When $\left(  E,\left\Vert \cdot\right\Vert _{E}\right) $ is a Banach space we can define a norm on $\mathcal{C}^{p-var}\left(  [0,T];E\right) $ via $$\begin{aligned}
\left\Vert f\right\Vert _{\mathcal{V}^{p};[0,T]}:=\left\Vert f\right\Vert_{p-var;[0,T]}+\sup_{t \in [0, T]}\left\Vert f_{t}\right\Vert_{E}.\end{aligned}$$

We will also need the notion of $p$-variation for two-parameter functions. Thus, given $f:[0,T]^{2} \rightarrow E$, we say $f$ is of finite 2D $p$-variation if $$\begin{aligned}
\left\Vert f\right\Vert _{p-var;[0,T]^{2}}:=\sup_{\pi}\left(  \sum_{i,j} \left\Vert f
\begin{pmatrix}
u_{i},u_{i+1}\\
v_{j},v_{j+1}
\end{pmatrix} \right\Vert_{E}^{p}\right)  ^{\frac{1}{p}}<\infty,\end{aligned}$$ where $\pi=\left\{  \left(  u_{i},v_{j}\right)  \right\}  $ is a partition of $[0,T]^{2}$, and the rectangular increment is given by $$\begin{aligned}
 \label{rectInc}
f \begin{pmatrix}
u_{i},u_{i+1}\\
v_{j},v_{j+1}
\end{pmatrix}
:=f(u_{i},v_{j})+f(u_{i+1},v_{j+1})-f(u_{i},v_{j+1})-f(u_{i+1},v_{j}).\end{aligned}$$ On occasion, we will use the notation $$\begin{aligned}
f(\Delta_{i},v):=f(u_{i+1},v)-f(u_{i},v)\text{ and }f(u,\Delta_{j} ) := f(u,v_{j+1})-f(u,v_{j}).\end{aligned}$$

Let $f$ and $g$ be functions defined on $[0,T]^{2}.$ We say that the 2D Young-Stieltjes integral of $f$ with respect to $g$ exists if there exists a scalar $I(f,g)\in\mathbb{R}$ such that $$\begin{aligned}
 \label{quant1}
\lim_{\left\Vert \pi\right\Vert \rightarrow0}\left\vert \sum_{i,j}f\left(
u_{i},v_{j}\right) g
\begin{pmatrix}
u_{i} & u_{i+1}\\
v_{j} & v_{j+1}\end{pmatrix}
-I(f,g)\right\vert \rightarrow0,\end{aligned}$$ i.e. for each $\varepsilon > 0$, there exists a $\delta>0$ such that for all partitions $\pi=\{(u_{i},v_{j})\}$ of $[0,T]^{2}$ with $\left\Vert \pi\right\Vert <\delta$, the quantity on 