The Hardy-Littlewood-Sobolev inequality in my book,
Let $0<\alpha < d$, $1<p<q<\infty$, and $\frac{1}{q} +1 = \frac {1}{p} + \frac {\alpha}{d}$. Then $$ \lVert f \ast |\cdot|^{\alpha} \rVert_{L^{q}(\mathbb{R}^d )} \leq C_{p,q} \lVert f \rVert_{L^{p}(\mathbb{R}^d)}.$$
But I saw another one,
$$\int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \frac {f(x)g(y)}{|x-y|^{\alpha}}dxdy \leq C_{\alpha, p, d }\lVert f \rVert_{L^{p}(\mathbb{R}^d)}\lVert g \rVert_{L^{q}(\mathbb{R}^d)}. $$ for all $f \in L^p, g \in L^q,\quad 1<p,q< \infty,\quad \frac{1}{p} + \frac{1}{q} + \frac {\alpha}{d} = 2$ and $0<\alpha <d $.
I want to show bottom one but I am failing it.