I was assigned to do the following problem and have been stuck with not many ideas.
It is a classical result that if $f\in C^2_{c}(\mathbb{R}^n)$ then the following functions give a solution of the Poisson equaion $-\Delta u=f$,
$$u(x)=\int_{\mathbb{R}^n} \Phi(x-y)f(y)\,\,\text{d} y.,$$ where $\Phi(x)$ is the fundamental solution of Laplace's equation.
The problem I have to solve is the following:
Given a function $f\in C^2(\mathbb{R}^n)$ (not necessarily with compact support) that satisfies $|f(x)|\leq \frac{C}{|x|^{\gamma}}$ and $|Df(x)|\leq \frac{C}{|x|^{\gamma+1}}$ for $\gamma$ a positive number. Find conditions on $\gamma$ so that the above defined function $u(x)$ still solves the Poincare equation.
Any help will be greatly appreciated.