Weighted Hardy-Littlewood-Sobolev inequality in one dimesion states:
Let $1 < p, q < \infty, 0 < t < 1, a, b \in \mathbb{R}$ such that:
$$ -t \leq a + b \leq 0, \frac{1}{p} < a + 1, \frac{1}{q} < b + 1, \frac{1}{p} + \frac{1}{q} = a + b + t + 1 $$
Then there exists $C$ (depending on $a,b,p,q,t$) such that: $$ \int_{\mathbb{R}}\int_{\mathbb{R}} \frac{|x|^{a}|f(x)||y|^{b}|g(y)|}{|x-y|^{1-t} }dxdy \leq C\left\| f\right\| _{L^p(\mathbb{R})}\left\| g\right\| _{L^q(\mathbb{R})}.$$
The proof can probably be done by applying Marcinkiewicz interpolation. This is similar to the unweighted case, but the kernel is giving me more problems here, so some help is welcome.