How to show that  fundamental solution of Laplace equation  $\in L^2 $given $ f  \in L^2 $?  I need help with this homework 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.

 A: The integrability of $u$ is a consequence of the following Lemma. It appears in Jost, Partial differential equations.
Lemma. For $\mu \in (0,1]$ and $f \in L^1(\Omega)$, $\Omega \subset \mathbb{R}^d$, put
$$
(V_\mu f)(x)=\int_\Omega |x-y|^{d(\mu-1)} f(y)\, dy.
$$
Let $1 \leq p \leq q \leq \infty$,
$$
0 \leq \delta = \frac{1}{p}-\frac{1}{q} < \mu.
$$
The $V_\mu$ maps continuously $L^p(\Omega)$ into $L^q(\Omega)$, and
$$
\|V_\mu f\|_q \leq \left( \frac{1-\delta}{\mu-\delta}\right)^{1-\delta} \omega_d^{1-\mu} |\Omega|^{\mu-\delta} \|f\|_p.
$$
Here $\omega_d$ is the volume of the unit ball of $\mathbb{R}^d$ and $|\Omega|$ is the Lebesgue measure of $\Omega$.
The proof of this lemma is a repeated application of the Hoelder inequality. I finally suspect that 3. requires an estimate of the decay of $u$ at infinity.
Probably you want to write 
$$\frac{1}{|x-y|} = \frac{1}{|x-y|}-\frac{1}{|x|}+\frac{1}{|x|}.
$$ 
Hence 
$$\int \frac{f(y)}{|x-y|}dy = 
\int \frac{f(y)}{|x|}dy + \int f(y) \left( \frac{1}{|x-y|}-\frac{1}{|x|} \right) dy.
$$ 
The first integral is zero because $\int f(y)dy=0$. Now you have to estimate the second integral when $|x|$ is large.
