Assume we have a Hida test function $\varphi\in (\mathcal S)$, and $y\in \mathcal S'(\mathbb R)$. Define the Gateaux directional derivative of $\varphi$ (in the direction of $y$) by:

$$D_y\varphi(x):=\sum_{n=1}^{\infty} n \langle :x ^{\otimes (n-1)}:,\langle y, f_n\rangle\rangle,$$

where the sequence of (symmetric) $f_n$'s is the element in the Fock space that corresponds to $\varphi$ (via the Ito-Segal isomorphism).

At this point I need to calculate the $S$-transform of this element, but I am having some problems (probably trivial stuff) with the calculations:

I have that by definition of the $S$-transform we have

$$S(D_y\varphi(x))(\xi)=\sum_{n=1}^{\infty} n\big\langle \langle y,f_n\rangle, \xi^{\otimes (n-1)}\big\rangle.$$

From this I should be able to obtain

$$\sum_{n=1}^{\infty} n\big\langle y\hat{\otimes} \xi^{\otimes (n-1)},f_n \big\rangle,$$

but this very last step is not clear to me and I haven't been able to find an explanation.

N.Obata justifies this step by using contraction tensors but prefer to avoid that formulation.

Could you please explain me this? I have the feeling that it's something straightforward but I haven't been able to get it.

Thanks in advance!

EDIT:

A colleague told me that this could be a consequence of the Kernel Theorem , but honestly I don't see how.

EDIT 2:

So this is very informal but if one thinks of the pairing $\langle \cdot,\cdot\rangle$ as the inner product in $(L^2)$ (which is not entirely correct since one of the element could be outside $(L^2)$ i.e. if $y\in S'(\mathbb R)\backslash L^2(\mathbb R)$) we can write that:

\begin{align} &\big\langle \langle y,f_n\rangle, \xi^{\otimes (n-1)}\big\rangle\\ =&\int_{\mathbb R^{n-1}} \int_{\mathbb R} f_n(t_1,\cdots,t_n)y(t_1)\xi^{\otimes (n-1)}(t_2,\cdots,t_n) dt_1d(t_2,\cdots,t_n)\\ =&\int_{\mathbb R^{n}}f_n(t_1,\cdots,t_n)y(t_1)\xi^{\otimes (n-1)}(t_2,\cdots,t_n) d(t_1,t_2,\cdots,t_n)\\ =&\langle y\otimes \xi^{\otimes (n-1)},f_n\rangle, \end{align}

But again this is informal and I don't know if I can justify all steps, what do you think?