**The question**


I consider the Laplacian $\Delta = \partial\_1^2 + \partial\_2^2 + \partial\_3^2$ in $\mathbb{R}^3$. By the "standard" fundamental solution of the Laplacian, I mean the function


$$ \displaystyle E(x) = C|x|^{-1} $$


where $C$ is some normalization constant. *I would like to know if one can construct a fundamental solution of the Laplacian that decays faster than any rational function at infinity*. I have an idea how one could maybe construct this, but I'm not sure if my idea is correct as the idea of a suspicious fundamental solution seems highly suspicious to me. In the following, I sketch my idea:


**An attempt at an answer**


I already know that one can construct a rapidly decaying parametrix $F$ by multiplying $\hat E \sim |x|^{-2}$ with a cutoff function $\chi$ supported in the complement of the unit ball and then taking the inverse Fourier transform. Then, one has $$x^\alpha F(x) = \int D^\alpha(\chi\hat E)(\xi)e^{ix\xi}\text{d}\xi $$
and the right hand side is absolutely convergent for large $\alpha$. This proves the decay at infinity.


I want to modify this construction in order to turn the parametrix $F$ into a proper fundamental solution. To do this, I want to use a simplified version of the construction by Hörmander (see "Distribution" by Duistermaat and Kolk, Theorem 17.11). Letting $\eta$ be a unit vector, I set


$$ G(x) = (2\pi)^{-n} \int \frac{1- \chi(\xi)}{2\pi i}\int\_{|z| = 100} \frac{e^{ix\cdot(\xi + z \eta)}}{|\xi + z\eta|^2}\frac{\text{d}z}{z}\text{d}\xi\,.$$


With Cauchy's integral theorem, one can show that $$\tilde E = F + G$$ is indeed a fundamental solution. The idea here is that we avoid the singularity at $0$ by taking a contour integral around it. It seems as if one can use the same trick that one used for $F$ to show the rapid decay of $G$. This suggest that one can indeed find a rapidly deacying fundamental solution. 



Are there fundamental solutions of the laplacian that decay rapidly?
No. You want the Fourier transform to satisfy $-|\xi|^2 \widehat{E}(\xi)=1$, so $\widehat{E}=-1/|\xi|^2 + \widehat{F}$, with $\textrm{supp}\:\widehat{F}=\{0\}$, but this says that $\widehat{F}$ is a linear combination of $\delta$ and its derivatives, so $F$ is a polynomial.


