Let $0< \alpha< n$, $1 < p < q < \infty$ and $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$. Then:
$ \left \| \int\_{\mathbb{R}^n} \frac{f(y)dy}{|x-y|^{n-\alpha} } \right\|\_{L^q(\mathbb{R}^n)}\leq$ $C\left\| f\right\| \_{L^p(\mathbb{R^n})}$.



Sobolev-type inequality.
This is the standard Hardy-Littlewood-Sobolev inequality(or the theorem of fractional integration).A more direct approach is write
$$
\int{f(x-y)|y|^{\alpha-n}dy}=\int\_{|y|<R}+\int\_{|y|\ge R}
$$ 
 For the second term on the RHS,using Holder inequality,and easy to see that it's dominated by $\|f\|\_{L^p}R^{-\frac{q}{n}}$.
 For the first term,one can use the majorizationgiven by the maximal function M,and to see that 
$$
|f\ast |y|^{\alpha-n}|(x)\leq C(M(f)(x)\cdot R^{\alpha}+\|f\|\_{L^p}\cdot R^{-\frac{q}{n}})
$$
 Choosing a proper constant R to make the two terms above be equal,and then the desired inequality hold by intergration(note that the maximal operator is bounded on $L^p$ for $1<p<\infty$).


