In Friedmans book there is analysis for the pde which is done via the fundamental solution. As I understand if we integrate that with initial data it gives us a solution of an equation. There are also results on regularity of the fundamental solution but I wonder if that has affect on the regularity of the solution itself? Is there a direct dependence? How smoothness of the fundamental solution affects the smoothness of a solution? thanks!
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Since the general solution of $Lf=g$ is the convolution of $g$ with the fundamental solution $\varphi$, we should expect that better regularity of $\varphi$ results in better regularity of $f=\varphi*g$. For example, if $\varphi\in L^p$ for some $p$, then $\varphi*g\in L^p$ for all $g\in L^1$, by Young's inequality. Of course, the more interesting question is the smoothness of solution rather than its integrability. Unfortunately, here the situation is complicated by the nature of $\varphi$: it is usually smooth except at one point where there is a singularity. One has to be careful estimating the contribution of the singularity to the solution. So, the relation between the smoothness of $\varphi$ and $\varphi*g$ cannot be described in broad terms that apply to all PDE.