Some text books on PDE (Evans, Gilbarg and Trudinger for instance) give a solution to Poisson's equation $-\Delta u = f$ on $\mathbb{R}^n$ as $u(x) = \int_{\mathbb{R}^n} \Phi(x - y)f(y)\; dy$, where $\Phi$ is the fundamental solution, and where $f$ is $C^2$ with compact support.
I think I've heard somewhere that $f$ having compact support is not necessary, only that $f$ decay fast enough in some sense at infinity.
My question is: does any one have a precise statement of this, or a reference?
For instance, in $\mathbb{R}^2$, would this
$$
|D_{ij}f(x)| \leq \frac{C}{(1 + |x|^2|)^2},
$$
for some constant $C$, be sufficiently fast decay to ensure that $-\Delta u = f$ on $\mathbb{R}^2$ even if $f$ does not have compact support?
Thank you for your advice!
EDIT: Or I am mistaken and compact support is necessary?