partitions $\pi:= \left\{  \left(  s_{i}, t_{j} \right)  \right\} $ of $[0, T]^{2}$.

Theorem 1 in [@fggr2016] states that if $R$ is of mixed $(1, \rho)$-variation, then $$\begin{aligned}
\left\|  h\right\| _{q-var; [0, T]} \leq\left\|  h \right\| _{\mathcal{H}^{d}}
\sqrt{\left\|  R\right\| _{(1, \rho)-var; [0, T]^{2}}},\end{aligned}$$ where $q = \frac{2\rho}{\rho+ 1}$. One can easily verify that this gives us $\frac{1}{p} + \frac{1}{q} > 1$ as long as $p < 4$.

The following condition collects the assumptions we impose on $X$, or equivalently $\mathbf{X}$.

\[newCond1\] Let $X$ be a continuous, centered Gaussian process in $\mathbb{R}^{d}$ with i.i.d. components, and assume that the covariance function satisfies

(a) $\left\|  R\right\| _{\rho-var; [0, T]^{2}} < \infty$ for some $\rho
    \in\left[ 1, 2 \right) $.

For $p \in[1, 4)$, let $\mathbf{X} \in\mathcal{C}^{0, p-var} \left( [0, T]; G^{\lfloor p \rfloor} \left( \mathbb{R}^{d}\right) \right) $ denote the geometric rough path constructed from the limit of the piecewise-linear approximations of $X$. Furthermore, assume that $\mathcal{H}^{d} \hookrightarrow\mathcal{C}^{q-var} \left( [0, T]; \mathbb{R}^{d} \right) $, where $q$ satisfies $\frac{1}{p} + \frac{1}{q} > 1$, i.e. for all $h \in\mathcal{H}^{d}$,

(b) $\left\|  h\right\| _{q-var; [0, T]} \leq C \left\|  h\right\|
    _{\mathcal{H}^{d}}$.

Later on, we will need to impose further conditions on the covariance function. For all $s, 