Given a Linear pde $L$, a distribution $u$ is said to be a fundamental solution if $Lu=\delta$ where $\delta$ is the Dirac delta distribution. A common example is the Newtonian potential which is the fundamental solution of the Laplacian. Are there any examples of differential operators of positive order with non-locally integrable fundamental solutions?
Short answer is yes. In 1999 Hörmander gave an example of a hypoelliptic operator where the fundamental solution is not locally integrable. He also proves, among other things, that no such counterexample exists in two dimensions.
This is the article: On local integrability of fundamental solutions by Hörmander Arkiv för Matematik Volume 37, Issue 1 , pp 121-140
The differential operator $1$ has $\delta$ as a fundamental solution, which is known to be non regular (not induced by a locally integrable function).
Also, the differential operator $x\partial/\partial x-1$ has $\delta$ as a fundamental solution, since $((x\partial/\partial x-1)\delta)(f)=\delta(-x\partial f/\partial x+f)=\delta(f)$. Are you looking for the constant coefficient case?