For $x\in\mathbb{R}^n$ we define the Newton potential as follows:$N(x) = \begin{cases} \frac{\log|x|}{2\pi}, & n=2 \\[10pt] \frac{|x|^{2-n}}{(2-n)\omega_n}, & n>2\end{cases}$
where $\omega_n$ denoted to the volume of the n-ball. Moreover, let $\chi_r$ denote to the characteristic function of the ball $B(0,r)$.
Now my lecture notes say (for $n>2$)
$\chi_r N \in L^1(\mathbb{R}^n)$ (or, more general, $L^p$, where $p<\frac{n}{n-2}$) and
$(1-\chi_r)N \in L^\infty(\mathbb{R}^n)$ (or, more general, $L^p$, where $p>\frac{n}{n-2}$)
Since $N \in L_\mathrm{loc}^1(\mathbb{R}^n)$, it follows immediately that $\chi_r N \in L^1(\mathbb{R}^n)$ for any $r>0$.
How do I see the other claims?