following lemma which characterizes the Malliavin derivative using the duality formula involving the Skorohod operator (also called divergence operator).

\[lemma:Sugita\] Let $\varepsilon>0$ and $1<p<+\infty$. Suppose that $F\in L^{1+\varepsilon}(\R)$ and assume that there exists $\mathcal{D}F$ in $L^p(H)$ such that: $$\E\left[ F\delta(G h)\right]=\E\left[ G \langle \mathcal{D}F,h\rangle_H\right],$$ for every $G\in \mathcal S$ and $h\in H$. Then, it holds that $F\in \D^{1,p}$, and $\mathcal{D}F=\nabla F, \; \P_0-a.s.$

We know (see e.g. [@Sugita Corollary 2.1]) that the result is true if $\mathcal{S}$ is replaced by $\mathcal{P}$. Let $G$ be in $\mathcal{P}$. By Lemma \[lemma:tec1\] there exists $(G^N)$ in $\mathcal{S}$ such that $G^N$ approximates $G$ in $\mathbb{D}^{1,p}$. Let $h$ in $H$. For any $N\geq 1$, we have $$\begin{aligned}
\E[F \delta(Gh)] &= \E[F (G W(h)-\langle \nabla G,h\rangle_H)]\\ 
&= \E[F (G^N W(h)-\langle \nabla G^N,h\rangle_H)] - \E[(G^N-G) F W(h)- F \langle \nabla (G^N-G),h\rangle_H]\\
&= \E[F \delta(G^Nh)] - \E[(G^N-G) F W(h)- F \langle \nabla (G^N-G),h\rangle_H]\\
&= \E[G^N \langle \mathcal{D} F, h\rangle_H] - \E[(G^N-G) F W(h)- F \langle \nabla (G^N-G),h\rangle_H].\end{aligned}$$ Furthermore, by Lemma \[lemma:tec1\] $$|\E[(G^N-G) F W(h)]| \leq \E[|F W(h)|^p]^{1/p} \E[|G^N-G|^{\bar{p}}]^{1/\bar{p}} \underset{N\to +\infty}{\longrightarrow} 0,$$ with $1<p<1+\varepsilon$ and where $\bar p$ is the conjugate of $p$, and $$|\E[F \langle \nabla (G^N-G),h\rangle_H]|\ \leq \E[|F|^p]^{1/p} \E[\|\nabla (G^N-G)\|_H^{\bar p}]^{1/\bar p} \|h\|_H \underset{N\to +\infty}{\longrightarrow} 0,$$ by 