the Malliavin sense and the derivatives up to order $k$ have finite moments of order $p$. If $F\in\mathbb{D}_{k,p}(V)$, we denote by $D^kF$ the $k$th Malliavin derivative of $F$, which is a random variable taking its values in the space $L^p(\Omega;H^{\otimes k}\otimes V)$. We write $DF$ instead of $D^1F$ for short. We set $\mathbb{D}_{k,\infty}(V)=\bigcap_{p=1}^\infty\mathbb{D}_{k,p}(V)$. If $V=\mathbb{R}$, we simply write $\mathbb{D}_{k,p}(V)$ as $\mathbb{D}_{k,p}$.

For a $d$-dimensional random vector $F\in\mathbb{D}_{k,p}(\mathbb{R}^d)$, we identify the $k$th Malliavin derivative $D^kF$ of $F$ as the $(H^{\otimes k})^d$-valued random variable $(D^kF^1,\dots,D^kF^d)$ . Similarly, for a $d\times d'$ matrix valued random variable $F\in\mathbb{D}_{k,p}(\mathbb{R}^{d\times d'})$, we identify $D^kF$ as the $(H^{\otimes k})^{d\times d'}$-valued random variable $(D^kF^{ij})_{(i,j)\in[d]\times[d']}$.

For a positive integer $q$, we denote by $\delta^q$ the $q$-th multiple Skorohod integral, which is the adjoint operator of the densely defined operator $L^2(\Omega)\supset\mathbb{D}_{q,2}\ni F\mapsto D^qF\in L^2(\Omega;H^{\otimes q})$. That is, the domain $\operatorname{Dom}(\delta^q)$ of $\delta^q$ is defined as the set of all $H^{\otimes q}$-valued random variables $u$ such that there is a constant $C>0$ satisfying $|E[\langle u,D^qF\rangle_{H^{\otimes q}}]|\leq C\|F\|_2$ for all $F\in\mathbb{D}_{q,2}$, and the following duality formula holds for any $u\in\operatorname{Dom}(\delta^q)$ and $F\in\mathbb{D}_{q,2}$: $$E[F\delta^q(u)]=E[\langle u,D^qF\rangle_{H^{\otimes q}}].$$

Multi-indices
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This subsection collects some notation related to multi-indices.

Let $q$ be a positive integer. We 