I know that people think about weak solutions to PDEs by turning a differential equation into an integral equation. Have people studied any analogue to this where we *start* with an integral equation, defined by a distribution rather than a vector field?

For instance, let $v$ be a distribution on $\mathbb{R}$ defined by $\delta_0 + 1$. I can write down an integral equation

$$\int_{-\infty}^{\infty} x(t) \omega'(t)\;\mathrm{d}t = \int_{-\infty}^{\infty} v(t) \omega(t)\;\mathrm{d}t$$

where $\omega$ ranges over all compactly-supported smooth functions on $\mathbb{R}$.

I believe that the function

$$x(t) = \begin{cases} t + C & \text{if $t < 0$} \\ t + C + 1 & \text{otherwise} \end{cases}$$

is then a solution to this integral equation.

If this works in simple cases, I'd like to generalize, as discussed [here](https://www.localcharts.org/t/differential-equations-with-jumps/16830), but it was recommended that I see if anyone had thought about the simple case first.