Lemma \[lemma:tec1\] again. Hence, $$\begin{aligned}
\E[F\delta(Gh)] &= \lim_{N\to+\infty}  \E[G^N \langle \mathcal{D} F, h\rangle_H]\\
&=\E[G \langle \mathcal{D} F, h\rangle_H] + \lim_{N\to+\infty}  \E[(G^N-G) \langle \mathcal{D} F, h\rangle_H],\end{aligned}$$ and $$\lim_{N\to+\infty} |\E[(G^N-G) \langle \mathcal{D} F, h\rangle_H]| \leq \lim_{N\to+\infty} \E[|G^N-G|^p]^{1/p} \E[\|\mathcal{D}F\|_H^{\bar p}]^{1/\bar p} \|h\|_H \underset{N\to +\infty}{\longrightarrow} 0.$$ Thus we have proved that for any $G$ in $\mathcal{P}$ and for any $h$ in $H$, $$\E[F\delta(Gh)] = \E[G \langle \mathcal{D} F, h\rangle_H],$$ which gives the result by [@Sugita Corollary 2.1].

We now prove the following lemma for the Malliavin differentiability of a given random variable.

\[lemma:D12nec\] Let $p>1$. Let $F$ be in $\D^{1,p}$. Then, for any $q$ in $[1,p)$ and for any $h$ in $H$, $$\frac{F{\circ \tau_{\varepsilon h}}-F}{\varepsilon}\underset{\varepsilon \to 0}{\longrightarrow} \langle \nabla F, h\rangle_H \textrm{ in } L^q({\mathbb{R}}).$$

Fix $q$ in $[1,p)$, $h$ in $H$ and $\eta>0$ such that $q+\eta<p$. We know from [@Sugita Theorem 3.1] that since $F$ is in $\D^{1,p}$, $F$ is (SGD), (RAC), and Relation holds true. We thus have using Jensen’s Inequality $$\begin{aligned}
\E\left[\left|\varepsilon^{-1} (F {\circ \tau_{\varepsilon h}}-F) \right|^{q+\eta}\right]&=\E\left[\varepsilon^{-(q+\eta)} \left| \int_0^\varepsilon \langle \nabla F {\circ \tau_{s h}}, h\rangle_H ds \right|^{q+\eta}\right]\\
&\leq\varepsilon^{-1} \E\left[\int_0^\varepsilon \left|\langle \nabla F {\circ \tau_{s h}}, h\rangle_H \right|^{q+\eta} ds\right]\\
&=\varepsilon^{-1} \int_0^\varepsilon \E\left[\left|\langle \nabla F, h\rangle_H \right|^{q+\eta} {\circ \tau_{s h}}\right] ds\\
&=\varepsilon^{-1} 