Fundamental solution of the wave operator What is the explicit formula for a fundamental solution of the wave operator, where the space variable is in $\mathbb{R}^n$ with $n>1$? Thanks.
The operator I'm talking about is 
$$L=\partial_{tt}-\Delta_x$$ 
and for fundamental solution I mean a distribution $E$ (temperate) which satisfies $LE=\delta$ where $\delta$ is the Dirac distribution. 
For $n=1$ one of such $E$ is $E(x,t)=(1/2)H(t)H(t^2-x^2)$ where $H$ is the Heaviside function.
I'm looking for expressions in higher dimensions.
 A: [Some of the links in the comments are dead.]
Fundamental solutions of the wave equations are truly singular, and before distributions it was almost impossible to describe. If you don't care about the constant in front, it can be described by repeated (distributional) derivatives of (regular) functions.
For n = odd, let m = (n-1)/2 and
$$ E = \square^m H(t-|x|) $$ 
but for n = even, with m = (n-2)/2
$$ E = \square^m \frac{H(t-|x|)}{\sqrt{t^2-|x|^2}} $$
Some remarks: In the first case, $E$ is supported on the surface of the (future) cone, and you can think of it as some higher derivative of the "$\delta$-function" along the surface. For the second case, the distribution is regular in the interior of the cone (and is $(t^2-x^2)^{-\frac{1}{2}-m}$ there); but this blows up along the boundary. What the distribution above tells you is how to "remove" the infinity. This is precisely what inspired Hadamard's theory of partie finie, which in turn was one of the motivations for Schwartz's distribution theory.
A: What you are calling $E$ here, sounds like the Green's function of the wave operator....look at the bottom of the wiki page D'Alembert operator for the explicit form.
A: Higher dimensional cases will follow from Poisson's and Kirchhoff's formulas.
