$($RAC$)$ and $($SGD$)$, for any $p>1$, see Lemma 15.89. Notice that [@janson] also obtained a similar characterization for $p=1$ $($see Lemma 15.71$)$. However, as stated in Remark 4 of [@Sugita], the identification of the Kusuoka-Stroock and Shigekawa spaces when $p=1$ is still an open result, so that we never consider the case $p=1$ in this paper.

The main result of this section is the following theorem whose proof is postponed to the end of the section.

\[th:newcarD\] Let $p>1$ and $F\in L^p({\mathbb{R}})$. The following properties are equivalent

-   $F$ belongs to $\D^{1,p}$.

-   There exists $\mathcal{D}F$ in $L^p(H)$ such that for any $h$ in $H$ and any $q\in [1,p)$ $$\lim\limits_{\varepsilon\to 0} \E\left[ {\left| \frac{F\circ \tau_{\varepsilon h}-F}{\varepsilon}-\langle \mathcal{D}F, h\rangle_H\right|}^q\right]=0.$$

-   There exists $\mathcal{D}F$ in $L^p(H)$ and there exists $q\in [1,p)$ such that for any $h$ in $H$ $$\lim\limits_{\varepsilon\to 0} \E\left[ {\left| \frac{F\circ \tau_{\varepsilon h}-F}{\varepsilon}-\langle \mathcal{D}F, h\rangle_H\right|}^q\right]=0.$$

-   There exists $\mathcal{D}F$ in $L^p(H)$ such that for any $h$ in $H$ $$\lim\limits_{\varepsilon\to 0} \E\left[ {\left| \frac{F\circ \tau_{\varepsilon h}-F}{\varepsilon}-\langle \mathcal{D}F, h\rangle_H\right|}\right]=0.$$

In that case, $\mathcal D F= \nabla F$.

The implication $(ii) \Rightarrow (i)$ when $q=p=2$ already appears in [@bog] $($see 8.11.3$)$. This is of course contained in our result.

We now give the 