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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?

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  • $\begingroup$ I have added the functional-analysis tag. $\endgroup$ Feb 26, 2014 at 8:14
  • $\begingroup$ I disagree with the statement that the fundamental solutions constructed using Hörmander's staircase are "bad". On the contrary, they have optimal regularity when measured on a Sobolev-type scale. Tempered fundamental solutions have their disadvantages such lack of continuous dependence on the operator. Think of $\partial_x+a$ and its tempered fundamental solutions when $a$ is near the imaginary axis for example. $\endgroup$
    – user80744
    Mar 2, 2014 at 14:19

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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

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Another approach uses Bernstein-Sato polynomials. If $P(s,\partial)$ is a differential operator with polynomial coefficients and $f(x)\in\mathbb{C}[x_1,\ldots,x_n]$ then the Bernstein-Sato polynomial for $P$ and $f$ satisfies $$P(s,\partial)f(x)^{s+1} = b(s) f(x)^s$$ If $f$ has only nonnegative values on $\mathbb{R}^n$, this can be used to define an meromorphic continuation of the distribution-valued, holomorphic function $\{Re(s)>0\}\to\mathcal{S}'(\mathbb{R}^n), s\mapsto f(x)^s$ and in particular a distributional inverse for $f(x)$. (If $f$ doesn't have nonnegative values on $\mathbb{R}^n$, $f(x)\overline{f(x)}$ does and that can be used instead)

Distributionally inverting polynomials is by Fourier transform equivalent to finding Green functions for constant-coefficient PDEs

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  • $\begingroup$ Yes, very good. The book by Coutinho on D-modules includes a very intelligible account of this particular thing, among others. $\endgroup$ Feb 26, 2014 at 0:24
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If you are interested in explicit fundamental solutions for pde's, you should take a look at the supremely interesting work of the Innsbruck group (P. Wagner and N. Ortner). Detailed references can be found via Google. For starters there is their "A survey on explicit representation formulae for fundamental solutions of linear partial differential operators" in Acta Applicandae Math. 47 (1997) 101-124.

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  • $\begingroup$ Yes, Peter Wagner's proof of the Malgrange-Ehrenpreis is superb. However, it neither gives good regularity (like Hörmander's construction) nor a tempered fundamental solution. $\endgroup$ Feb 26, 2014 at 8:11

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