After my previous post I got curious about the following very simple question (which I don't seem to find the answer). Given a tempered distribution $K \in \mathcal{S}'(\mathbb{R}^{n_{1}+\cdots+n_{N}})$, does if follow that: \begin{eqnarray} K(\varphi_{1}\otimes\cdots\otimes\varphi_{N}) = \int k(x_{1},...,x_{N})\varphi(x_{1})\cdots\varphi_{N}(x_{N})dx_{1}\cdots dx_{N}\tag{1}\label{1} \end{eqnarray} for some integral kernel $k$ and $\varphi_{j}\in \mathcal{S}(\mathbb{R}^{n_{j}})$? Here $(\varphi_{1}\otimes \cdots \otimes \varphi_{N})(x_{1},...,x_{N}) := \varphi(x_{1})\cdots\varphi_{N}(x_{n})$.

After my previous post I got curious about the following very simple question (which I don't seem to find the answer). Given a tempered distribution $K \in \mathcal{S}'(\mathbb{R}^{n_{1}+\cdots+n_{N}})$, does if follow that: \begin{eqnarray} K(\varphi_{1}\otimes\cdots\otimes\varphi_{N}) = \int k(x_{1},...,x_{N})\varphi_1(x_{1})\cdots\varphi_{N}(x_{N})dx_{1}\cdots dx_{N}\tag{1}\label{1} \end{eqnarray} for some integral kernel $k$ and $\varphi_{j}\in \mathcal{S}(\mathbb{R}^{n_{j}})$? Here $$ (\varphi_{1}\otimes \cdots \otimes \varphi_{N})(x_{1},...,x_{N}) := \varphi_1(x_{1})\cdots\varphi_{N}(x_{n}). $$