\int_0^\varepsilon \E\left[\left|\langle \nabla F, h\rangle_H \right|^{q+\eta} \mathcal{E}\left(s \int_0^T \dot{h}_r \cdot dW_r\right) \right]ds\\
&\leq \E\left[\left|\langle \nabla F, h\rangle_H \right|^p\right]^{\frac{q+\eta}{p}} \sup_{t\in (0,1)} \E\left[\left|\mathcal{E}\left(t \int_0^T \dot{h}_r \cdot dW_r\right)\right|^{\frac{p}{p-q-\eta}} \right]^{\frac{p-q-\eta}{p}}\\
&<+\infty.\end{aligned}$$ Hence by de La Vallée Poussin Criterion, we deduce that the family of random variables $\left(\left|\varepsilon^{-1}(F{\circ \tau_{\varepsilon h}} -F)\right|^q\right)_{\varepsilon \in (0,1)}$ is uniformly integrable which together with the convergence in probability gives the result.

Note that the conclusion of the previous Lemma may fail for $q=p$[^7]

We can now proceed with the proof of Theorem \[th:newcarD\].

**Proof of Theorem \[th:newcarD\].** From Lemma \[lemma:D12nec\] we have $(i)\Rightarrow (ii)$ and of course $(ii) \Rightarrow (iii)\Rightarrow (iv)$. We turn to $(iv)\Rightarrow (i)$. Let $F$ be such that there exists $\mathcal{D}F$ in $L^p(H)$ such that $$\lim\limits_{\varepsilon \to 0} \E\left[ {\left| \frac{F\circ \tau_{\varepsilon h}-F}{\varepsilon}-\langle \mathcal{D}F, h\rangle_H\right|}\right]=0.$$ The proof consists in applying Lemma \[lemma:Sugita\] by proving the duality relationship $$\label{eq:proofdual}
\E[F \delta(G h)] = \E\left[G \langle \mathcal{D}F, h\rangle_H \right], \ G\in \mathcal S, \;h \in H.$$ By Lemma \[lemma:SugitabisStep1Step1\] (in the Appendix) with $\varepsilon=0$, $$\begin{aligned}
\label{eq:temp1}
\E\left[ F \delta(G h) \right]= &\ \frac{d}{d\varepsilon }\E\left[ F{\circ \tau_{\varepsilon h}} \, G\right]_{\vert \varepsilon =0}\nonumber\\
=&\ \lim\limits_{\eta \to 0}\frac1\eta \E\left[(F{\circ \tau_{\eta h}}-F) G \right]\nonumber\\
=&\ \lim\limits_{\eta \to 0}\E\left[\left(\frac{F\circ \tau_{\eta h}-F}{\eta}- \langle \mathcal{D}F,h \rangle_H 