After a brief introduction to distribution theory, I have been given the definition of fundamental solution for an operator $\mathcal{L}$ as follows:
Given a domain $\Omega\subset\mathbb{R}^N$ and a point $y\in\Omega$, it is said that $K(x,y)$ is a fundamental solution of the differential operator $\mathcal{L}$ in $\Omega$ with pole $y$ when $$\mathcal{L}K(\cdot,y)=\delta_y\text{ in }\mathcal{D}'(\Omega)$$
we are trying to solve the equation $$\Delta u(x)=f(x)\text{ for }x\in\Omega$$ and the notes says that the following function should be a solution, here is where I am having troubles to understand $$u(y)=\int_{\Omega}K(x,y)f(x)dx$$ My problem is that, by defintion, the identity $$\varphi(y)=\int_{\Omega}K(x,y)\Delta\varphi(x)dx$$ only holds for every test functions, so my concrete question are:
1- Aren't we assuming the solution $u$ is a test function? Which would mean that it is $C^\infty$ instead of just $C^2$ as required by the equation.
2- This might be almost the same as the first question but, all we know is $f\in C(\Omega)$, so (I think) it can't be the Laplacian for any test function, and again the previous equation is verfied only by those.