continuously embedded into $q$-variation spaces, see e.g. [@cfv2009], [@fggr2016]. By combining these results with Young’s inequality, one can then say e.g. that $$\begin{aligned}
     \label{embed}
    \left\vert D_{g}Y_{t}\right\vert \lesssim\left\vert g\right\vert_{q\text{-var}}\lesssim\left\vert g\right\vert _{\mathcal{H}^{d}},\end{aligned}$$ and these arguments can be generalized to higher order directional derivatives, allowing one control over the Hilbert-Schmidt norm of the Malliavin derivative; see [@inahama2014]. Note however, that quality is lost in by use of the embedding. For the proof in (ii) we need subtler estimates on the higher order derivatives of the form $$\begin{aligned}
     \label{high o}
    \left\vert D_{g_{i},....,g_{n}}^{n}Y_{t}\right\vert \leq C_{n}\left(\mathbf{X}\right)
    {\textstyle\prod\limits_{j=1}^{n}} \left\vert g_{j}\right\vert _{q\text{-var}}.\end{aligned}$$ High order derivatives here complicate matters since derivatives of order $2$ and higher are no longer representable as Young integrals as in ; instead genuine rough integrals appear. Much of the work underpinning point (ii) goes into deriving closed-form expressions for these high-order derivatives and then estimating them so as to arrive at . We must also pay careful attention to the random variable $C_{n}\left(  \mathbf{X}\right)$ in which, for our application, must have finite positive moments of all orders. The first half of Section \[Sko aug\] is devoted to this material.

The culmination of the these arguments is presented in Section \[corr formula\], where we give 