the left of is less than $\varepsilon$. In this case, we use $\int_{[0,T]^{2}}f\,\mathrm{d}g$ to denote $I(f,g)$, or $\int_{[s,t]\times\lbrack u,v]}f\,\mathrm{d}g$ whenever we restrict ourselves to any particular subset $[s,t]\times\lbrack u,v]$ of $[0,T]^{2}$.

We say that $f\in\mathcal{C}^{p-var}([s,t]\times\lbrack u,v])$ and $g\in\mathcal{C}^{q-var}([s,t]\times\lbrack u,v])$ have complementary regularity if $p^{-1}+q^{-1}>1$.

The significance of this definition lies in the following theorem (see [@towghi2002], [@fv2010a]), which gives the existence of the Young-Stieltjes integral and Young’s inequality in two dimensions; see [@lcl2006], [@fh2014], [@fv2010b] for the one-dimensional version.

\[2Dintegral\] Let $f \in V^{p-var}([s, t] \times[u, v])$ and $g \in V^{q-var}([s, t] \times[u, v])$ have complementary regularity. We also assume that $f(s, \cdot)$ and $f(\cdot, u)$ have finite p-variation, and that $f$ and $g$ have no common discontinuities. Then the 2D Young-Stieltjes integral exists and the following Young’s inequality holds; $$\begin{aligned}
 \label{2DYoungIneq}
\begin{split}
\left|  \int_{[s,t] \times[u,v]} f \, \mathrm{d} g\right|  \quad\leq C_{p, q}
\, {\left\vert \kern-0.25ex\left\vert \kern-0.25ex\left\vert f \right\vert
\kern-0.25ex\right\vert \kern-0.25ex\right\vert } \left\|  g\right\| _{q-var,
[s,t] \times[u,v]},
\end{split}\end{aligned}$$ where $$\begin{aligned}
{\left\vert \kern-0.25ex\left\vert \kern-0.25ex\left\vert f \right\vert
\kern-0.25ex\right\vert \kern-0.25ex\right\vert } = \left|  f(s, u)\right|  +
\left\|  f(s, \cdot)\right\| _{p-var; [u, v]} + \left\|  f(\cdot, u)\right\|
_{p-var; [s, t]} + \left\|  f\right\| _{p-var, [s,t] \times[u,v]}.\end{aligned}$$

Gaussian rough paths
--------------------

We will work with 