dual space to $H^{r}\left(  \mathbb{R}^{d},\mathbb{R}^{d}\right)
$, or equivalently the space of all vector valued distributions $\varphi$ such that $\int_{\mathbb{R}^{d}}\left(  1+\left|  x\right|  ^{2}\right)  ^{-r}\left|
\widehat{\varphi}\left(  x\right)  \right|  ^{2}dx<\infty$, $\widehat{\varphi
}$ being the Fourier transform of $\varphi$. Assume first that $d=1$. The case $d\geq 2$ will be treated later.

We will recall the following lemma (see [@FT], Lemma 1).

\[ahat\] Let $s=(s_{1},.., s_{N})\in [0,1]^{N}$ be fixed. The Fourier transform of the function $$x\to a_{m} ^{x} (s) : \mathbb{R} \to \mathbb{R}$$ is denoted by $a_{m}^{\hat{x}}(s)$ and it is given by $$a_{m}^{\hat{x}}(s)= e^{-\frac{x^{2}}{2} \vert s\vert } \frac{ (-i) ^{m} x^{m} }{m!}$$ where $\vert s\vert =s_{1}...s_{N}$.

As a consequence, the Fourier transform with respect to $x$, denoted in the sequel $\Delta (\widehat{x}-W_{u_{1}, v_{2}}) $, of the random distribution $x\in \mathbb{R} \to \Delta (x-W_{u_{1}, v_{2}}) $ is given by $$\Delta (\widehat{x}-W_{u_{1}, v_{2}})=\sum_{m} a_{m}^{\hat{x}}(u_{1}, v_{2}) I_{m} \left( 1_{[0,u_{1}]\times [0, v_{2}]} ^{\otimes m} \right)= e^{-ixW_{u_{1}, v_{2}} }.$$

\[pp1\] The Fourier transform of the function $$x\to \xi (x)$$ is given by $$\hat{\xi} (x) = \delta _{(u_{1},v_{1}), (u_{2},v_{2})}^{(2)} \left(   e^{-ixW_{u_{1}, v_{2}}}1_{[0, u_{1}] \times [0, v_{2}] } \right).$$

[**Proof:** ]{}Using formula (\[xi-sim\]) and the expression of the Fourier transform of $x\to a_{m}^{x}$ from Lemma \[ahat\], we get (to justify the interchange 