Using *Hardy-Littlewood-Sobolev* inequality, we can prove that:
$$\left| \int_0^1\int_0^1 |x-y|^{-\frac{1}{2}} f(x)f(y) \mathrm{d}x \mathrm{d}y \right| \leq C \left\| f \right\|_{L^{4/3}(0,1)}^2 \leq C \left\| f \right\|_{L^{2}(0,1)}^2.$$

However, the left-hand side looks very similar to the singular integrals used to define fractional Sobolev norms (actually, *Gagliardo* seminorms), maybe for a negative fractional index. Thus, I would expect that we could actually prove something like:
$$\left| \int_0^1\int_0^1 |x-y|^{-\frac{1}{2}} f(x)f(y) \mathrm{d}x \mathrm{d}y \right| \leq C \left\| f \right\|_{H^{-1/4}(0,1)}^2,$$
where $H^{-\frac{1}{4}}(0,1)$ is defined either as the dual of the usual fractional Hilbert space $H^{\frac{1}{4}}(0,1)$ or using *Fourier* series expansions for functions on $(0,1)$.

**Is this second inequality true? If so, where should I be looking for its proof?**