Hi, I am really struggling with this question.

The question is :

Let $f:R^3\to R$ and $f\in L^2(R^3)$. $f$ is supported on a ball of radius 1/2 centred at origin. Let $u$ be the solution to $\Delta u=f$ , where $ u $ is given by $u(x)= \frac{1}{4\pi}\int_{R^3}\frac{1}{|x-y|}f(y)\,dy$.

- Show that $L^2$ norm of u in the unit ball of radius 1, centred at origin, is bounded by C$||f||_{L^2}$, where C is a constant independent of f.
- Show that $u$ is $C^\infty$ outside the unit ball centred at origin.
- Suppose that $\int_{R^3}f(y)dy = 0$ , show $u\in L^2(R^3)$. (Consider how an good approximation it is to replace $\frac{1}{|x-y|}$ by $\frac{1}{|x|}$ for $|x|$ large.

$(1)$ seems straightforward since $f$ is compactly supported. For (2), I managed to show $\frac{1}{|x-y|}$ can be approximated by a sequence of smooth function say $\delta_n$ such that $u(x)= \int \delta_n(x-y)f(y)dy$ . But no success.

Could anyone here help me with this problem? Thanks in advance.