Let $U\subset\mathbb{R}^{d}$ be an open set and consider $X=\mathbb{R}\times U$. Now, lets consider a smooth (regular) kernel $k_{A}\in C^{\infty}(X\times X)$ and corresponding continuous operator $A:C^{\infty}_{c}(X)\to C^{\infty}(X)$ defined by $$A\varphi(x)=\int_{X}d^{d+1}y\,\,k_{A}(x,y)\varphi(y)$$ Now, writing $x=(t,\vec{x})$ and $y=(\tau,\vec{y})$, I can define a family of operator $A_{t,\tau}:C^{\infty}_{c}(U)\to C^{\infty}(U)$ by $$A\varphi(t,\vec{x})=\int_{\mathbb{R}}d\tau\,\underbrace{\int_{U}d^{d}\vec{y}\,\,k_{A}((t,\vec{x}),(\tau,\vec{y}))\varphi(\tau,\vec{y})}_{=:(A_{t,\tau}\varphi_{\tau})(\vec{y})}=\int_{\mathbb{R}}d\tau\,\, A_{t,\tau}\varphi_{\tau}(\vec{x})$$ where $\varphi_{\tau}(\cdot):=\varphi(\tau,\cdot)$, obviously. The operator $A_{t,\tau}$ are something like a time-kernel.
Question: Is there a way to make this notion of time-kernels precise also for non-regular kernels?
I consider (continuous) operator of the form $A:C^{\infty}_{c}(X)\to C^{\infty}(X)$. By the kernel theorem of Schwartz, there is a (semi-regular, i.e. smooth in x) kernel $k_{A}\in\mathcal{D}^{\prime}(X\times X)$. Now, informally, I would like to define a time-kernel, i.e. operators of the form $A_{t,\tau}:C^{\infty}_{c}(U)\to C^{\infty}(U)$ such that informally, I have something like above, i.e. $$A\varphi(t,\vec{x})=\int_{\mathbb{R}}d\tau\,\, A_{t,\tau}\varphi_{\tau}(\vec{x})$$ Of course, for honest distributions, the integral is just a formal notation, so I am a bit unsure how to even define the notion of time-kernels in this case. Nevertheless, I have seen people using the time-kernel notation in various different contexts, also for honest distributions (like the Green's functions in PDE, see for example beginning of Section 11.5 here etc). Also in the context of pseudodifferential operators I have seen this notion appearing.
Any insight into this is appreciated.
