B^H_{\cdot}\right]}\right\rVert}_{q-var; [0, s]}
\leq {\left\lvert{\mathbb{E} \left[ B^H_{s,t} B^H_s\right]}\right\rvert}
\leq {\left\lvertt - s\right\rvert}^{2H},\end{aligned}$$ where we note that ${\left\lvert{\mathbb{E} \left[ B^H_{s,t} B^H_u\right]}\right\rvert} < {\left\lvertt - s\right\rvert}^{2H}$ for all $u$ if $H \leq \frac{1}{2}$, cf. Lemma 5 in [@nnt2010]. The bound for ${\left\lVert{\mathbb{E} \left[ B^H_{s,t} B^H_{\cdot}\right]}\right\rVert}_{q-var; [t, T]}$ is shown in the same manner when we again exploit the fact that the disjoint increments have the same sign.

Given a Banach space $E$ and a kernel $K$ satisfying Condition \[amnCond\] for some $\alpha\in\left[  0,\frac{1}{4}\right)  $, we introduce the linear operator $\mathcal{K}^{\ast}$ (see [@amn2001], [@dfond2005]) $$\begin{aligned}
 \label{kStarDefn}
\left(  \mathcal{K}^{\ast}\phi\right)  (s):=\phi\left(  s\right)
K(T,s)+\int_{s}^{T}\left[  \phi\left(  r\right)  -\phi\left(  s\right)
\right]  K(\mathrm{d}r,s),\end{aligned}$$ where the signed measure $K(\mathrm{d}r,s):=\frac{\partial K(r,s)}{\partial r}\,\mathrm{d}r$. The domain $D\left( \mathcal{K}^* \right)$ of $\mathcal{K}^*$ consists of measurable functions $\phi:\left[0, T \right] \rightarrow E$ for which the integral on the right-hand side exists for all $s$ in $\left[  0,T\right] $. In particular, if $\phi$ is a $\lambda$-Hölder continuous function in the norm of $E$ for some $\lambda > \alpha$, then $\phi \in D\left( \mathcal{K}^* \right) $ and $\mathcal{K}^* \phi$ is in $L^{2}([0,T];E)$. Note also that for any $a$ in $[0,T]$, $\phi\mathds{1}_{[0,a)}$ is in $D\left( \mathcal{K}^* \right) $ whenever $\phi$ is, and we have the identity $$\begin{aligned}
 \label{truncatedKStar}
