TITLE: Method to compute fundamental solutions which are distributions
QUESTION [7 upvotes]: The Malgrange-Ehrenpreis theorem tells us that there is a fundamental solution for any linear differential operator of constants coefficients. The original proof was not constructive (it was based on the Hahn-Banach theorem). Lars Hörmander first gave a method (Hörmander's staircase) to explicitely compute fundamental solutions. Nevertheless, the solutions his method provides are usually "too bad", for they are not even tempered distributions.
Therefore, a natural question arises: Is there a general method  to explicitely construct fundamental solutions which are actually tempered distributions?

REPLY [5 votes]: The fact that there always exists a temperate fundamental solution was proved by Hormander here in 1958. I don't know if this answers some part of your question. Concerning regularity, well, this is an interesting question (most of the classical theory of PDEs is concerned with it). 
Let me also mention a nice paper by 
Rosay who reproves Malgrange-Ehrenpreis in the $L^2$ setting with an elementary proof, but still non constructive