by the kernel as a consequence of the bounds above.

Now to prove that $$\begin{aligned}
{\left\lVertR(t, \cdot) - R (s, \cdot) \right\rVert}_{q-var; [0, T]}  \leq C \left| t - s\right|^{\frac{1}{\rho}},\end{aligned}$$ when $\frac{1}{4} < H \leq \frac{1}{2}$, we will adopt the method in [@fv2011], and find bounds for $$\begin{aligned}
{\left\lVert{\mathbb{E} \left[ B^H_{s,t} B^H_{\cdot}\right]}\right\rVert}_{q-var; [s, t]}, \quad
{\left\lVert{\mathbb{E} \left[ B^H_{s,t} B^H_{\cdot}\right]}\right\rVert}_{q-var; [0, s]}, \quad \mathrm{and} \;
{\left\lVert{\mathbb{E} \left[ B^H_{s,t} B^H_{\cdot}\right]}\right\rVert}_{q-var; [t, T]},\end{aligned}$$ for all $s, t$ in $[0, T]$. For the first quantity in the preceding line, we use the fact that when $H > \frac{1}{4}$, we have (see [@fggr2016]) $$\begin{aligned}
{\left\lVertR\right\rVert}_{1, \rho-var; [s,t]^2} \leq C {\left\lvertt - s\right\rvert}^{2H},\end{aligned}$$ and $$\begin{aligned}
{\left\lVert{\mathbb{E} \left[ B^H_{s,t} B^H_{\cdot}\right]}\right\rVert}_{q-var; [s, t]}
&\leq {\left\lVertR(t, \cdot) - R(s, \cdot)\right\rVert}_{\mathcal{H}} \sqrt{{\left\lVertR\right\rVert}_{1, \rho-var; [s,t]^2}} \\
&\leq C {\left\lvertt - s\right\rvert}^H {\left\lvertt - s\right\rvert}^H = C {\left\lvertt-s\right\rvert}^{2H}.\end{aligned}$$ Now let $\{r_i\}$ be a partition of $[0, s]$. Then $$\begin{aligned}
\sum_i \left( {\mathbb{E} \left[ B^H_{s,t} B^H_{r_i, r_{i+1}}\right]} \right)^q
&\leq \left( \sum_i {\left\lvert{\mathbb{E} \left[ B^H_{s,t} B^H_{r_i, r_{i+1}}\right]}\right\rvert} \right)^q \\
&= {\left\lvert{\mathbb{E} \left[ B^H_{s,t}B^H_s\right]} \right\rvert}^q\end{aligned}$$ since the disjoint increments of fractional Brownian motion have non-positive correlation when $H \leq \frac{1}{2}$. Taking the supremum over all partitions of $[0, s]$, we have $$\begin{aligned}
{\left\lVert{\mathbb{E} \left[ B^H_{s,t} 