It is known that the solution of
$$-\Delta u = f \quad \text{in } \mathbb{R}^3$$
can be represented through the volume potential
$$ Vf(x) = \int_{\mathbb{R}^3} E(x-y)f(y) \, dy, \quad x \in \mathbb{R}^3, $$
where $E(x-y)$ denotes the fundamental solution in dimension three, provided $f$ has some regularity. Whether we talk about classical or weak solutions, it is usually assumed that $f$ has compact support in $\mathbb{R}^3$. I was wondering: are there similar results available in the case $f \in L^p(\mathbb{R}^3)$ with a suitable decay behaviour at infinity ? For instance, if
$$ f(y) = O(|y|^{-3}) \quad \text{as } |x|\to \infty,$$
then $f(y)E(x-y)$ seems to decay well enough to ensure convergence of the integral. Nevertheless I could only find results that require $f$ to be compactly supported and that's also what the classical proofs exploit.