$ r'=\sqrt{|x-x'|^2+|y-y'|^2+|z-z'|^2}$
If for $x',y',z',t \in R$
For a continuous function $f(x,y,z,t)$ with partial derivatives that exist at every point and the function and the partial derivatives are absolutely integrable.
let: $$g(x,y,z,t)=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac {f(x',y',z',t-r')}{4\pi r'}dx'dy'dz',\ \ \text{s.t.}\ \ \nabla^2g= \frac {\partial^2g}{\partial x^2}+\frac {\partial^2g}{\partial y^2}+\frac {\partial^2g}{\partial z^2}$$
Under what sufficient conditions we have $\nabla^2g-\frac {\partial^2g}{\partial t^2}=-f$ ?
Remark: The above equation is known as the Inhomogeneous Wave Equation. I was only able to answer the question under some conditions and doesn't look pretty. I believe there are condition which are less restrictive. Here are the conditions I found:
DEFINITIONS:
$ d'=\sqrt{|x'|^2+|y'|^2+|z'|^2}$
$f_x=\frac {\partial f}{\partial x}$
$f_{xx}=\frac {\partial^2 f}{\partial x^2}$
$f_{xt}=\frac {\partial}{\partial x}\frac {\partial f}{\partial t}$
$w_x(x',y',z',t)=|f_x(x',y',z',t-d')|+|f_{xx}(x',y',z',t-d')|+|f_{xt}(x',y',z',t-d')|$
$w_y(x',y',z',t)=|f_y(x',y',z',t-d')|+|f_{yy}(x',y',z',t-d')|+|f_{yt}(x',y',z',t-d')|$
$w_z(x',y',z',t)=|f_z(x',y',z',t-d')|+|f_{zz}(x',y',z',t-d')|+|f_{zt}(x',y',z',t-d')|$
$w_t(x',y',z',t)=|f_t(x',y',z',t-d')|+|f_{tt}(x',y',z',t-d')|$
CONDITIONS:
Let a continuous function $f(x,y,z,t)$ have partial derivatives $f_x,f_y,f_z,f_t$ , $f_{xx},f_{yy},f_{zz}$, $f_{xt},f_{yt},f_{zt}$ exist at every point and the function and all the above partial derivatives are absolutely integrable. If for $x',y',z',t \in R$ there is a non negative Lebesgue integrable function $h(x',y',z')$ such that :
$f_{xt}=f_{tx},f_{yt}=f_{ty},f_{zt}=f_{tz}$
$w_x(x+x',y',z',t)< h(x',y',z')$ almost everywhere in $R^3$
$w_y(x',y+y',z',t)< h(x',y',z')$ almost everywhere in $R^3$
$w_z(x',y',z+z',t) < h(x',y',z')$ almost everywhere in $R^3$
$w_t(x',y',z',t) < h(x',y',z')$ almost everywhere in $R^3$
$\lim_{x \to \infty}\frac {\partial f}{\partial t} =\lim_{y \to \infty}\frac {\partial f}{\partial t}=\lim_{z \to \infty}\frac {\partial f}{\partial t} = 0$