Let $U\subset\mathbb{R}^{d}$ be an open subset and set $M:=I\times U$, where $I=(a,b)\subset\mathbb{R}$ is some open subset. Lets consider a linear operator $B:C^{\infty}_{c}(M)\to C^{\infty}(M)$ that is continuous w.r.t. the usual Fréchet topologies. Then, by the Schwartz kernel theorem, there is a kernel$^{1}$ $k_{B}\in\mathcal{D}^{\prime}(M\times M)$ such that $$\langle B\varphi,\psi\rangle_{M}=\langle k_{B},\psi\otimes\varphi\rangle_{M\times M}$$ for $\varphi,\psi\in C^{\infty}_{c}(M)$, where $\langle\cdot,\cdot\rangle$ denotes the pairing of distributions.
Now, lets suppose for a moment that $k_{B}\in C^{\infty}(M\times M)$, i.e. my operator $B$ is a smoothing operator. In this case, I can define a two-parameter family of operator $B_{t,\tau}:C^{\infty}_{c}(U)\to C^{\infty}(U)$ by $$(B_{t,\tau}\varphi)(\vec{x}):=\int_{U}k_{B}(t,\vec{x};\tau,\vec{y})\varphi(\vec{y})\,d\vec{y}$$ so that my operator $B$ can be written as $$(B\varphi)(t,\vec{x})=\int_{I}B_{t,\tau}\varphi(\tau,\vec{x})\, d\tau$$ Now, I would like to do something similar in the case, in which $k_{B}$ is an honest distribution, i.e. in which I cannot write the distributional pairing $\langle\cdot,\cdot\rangle_{M}$ as in integral.
Question: Does some sort of "time-kernel" always exist and if yes, how to define it rigorously without using integral notation?
In my case, in fact $k_{B}\in C^{\infty}(M)\hat{\otimes}\mathcal{D}^{\prime}(U)\subset\mathcal{D}^{\prime}(M\times M)$, since my operator $B$ has smooth range.
