n}$ are (uniquely determined) symmetric functions and $I_{0}(f_{0})=\mathbf{E}\left[  F\right]  $.

Let $L$ be the Ornstein-Uhlenbeck operator $$LF=-\sum_{n\geq 0} nI_{n}(f_{n})$$ if $F$ is given by (\[sum1\]).

For $p>1$ and $\alpha \in \mathbb{R}$ we introduce the Sobolev-Watanabe space $\mathbb{D}^{\alpha ,p }$ as the closure of the set of polynomial random variables with respect to the norm $$\Vert F\Vert _{\alpha , p} =\Vert (I -L) ^{\frac{\alpha }{2}} \Vert
_{L^{p} (\Omega )}$$ where $I$ represents the identity. In this way, a random variable $F$ as in (\[sum1\]) belongs $\mathbb{D}^{\alpha , 2}$ if and only if $$\label{d12}
\sum_{n\geq 0} (1+n) ^{\alpha } \Vert I_{n}(f_{n}) \Vert
_{L^{2}(\Omega)} ^{2} =\sum_{n\geq 0} (1+n) ^{\alpha }n! \Vert f_{n}
\Vert ^{2} _{{\cal{H}}^{\otimes n}}<\infty .$$ We denote by $D$ the Malliavin derivative operator that acts on smooth functions of the form $F=g(B(\varphi _{1}), \ldots , B(\varphi_{n}))$ ($g$ is a smooth function with compact support and $\varphi_{i} \in {{\cal{H}}}$) $$DF=\sum_{i=1}^{n}\frac{\partial g}{\partial x_{i}}(B(\varphi _{1}), \ldots , B(\varphi_{n}))\varphi_{i}.$$ The operator $D$ is continuous from $\mathbb{D} ^{\alpha , p} $ into $\mathbb{D} ^{\alpha , p} \left( {\cal{H}}\right).$ The adjoint of $D$ is denoted by $\delta $ and is called the divergence (or Skorohod) integral. It is a continuous operator from $\mathbb{D}
^{\alpha, p } \left( 