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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?

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One has, for $f,\,g\in \dot{H}^{-1/4}(\mathbb R)$, $$ \begin{aligned} \left|\int_{-\infty}^\infty\int_{-\infty}^\infty |x-y|^{-1/2}f(x) \,\overline{g(y)}\,dx dy\,\right| &= C_0 \left|\left((-\Delta)^{-1/4}f,g\right) \right| \\ & \leq C_0\|(-\Delta)^{-1/4}f\|_{\dot{H}^{1/4}}\|g\|_{\dot{H}^{-1/4}} = C_0\|f\|_{\dot{H}^{-1/4}} \|g\|_{\dot{H}^{-1/4}}, \end{aligned} $$ where $(\;,\,)$ denotes the $L^2$ scalar product and $C_0>0$ is some constant. Now use the fact that $$ H^{-1/4}((0,1))= \{f\in \dot{H}^{-1/4}(\mathbb R)\mid \operatorname{supp}f\subseteq [0,1]\}. $$

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  • $\begingroup$ Thanks. This answer is great! What reference would you suggest for fractional Laplacian and homogeneous Sobolev spaces? As a subsidiary question, I was wondering if the opposite inequality could also be proved, maybe with an additional hypothesis on $f$? That is, my goal would be to prove that we also have: $$\left| \int_0^1\int_0^1 |x-y|^{-\frac{1}{2}} f(x)f(y) \mathrm{d}x \mathrm{d}y \right| \geq C \left\| f \right\|_{H^{-1/4}(0,1)}^2.$$ $\endgroup$ – F.M. Apr 14 '15 at 19:15
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    $\begingroup$ Indeed, $H^{-1/4}((0,1))$ is a closed subspace of $\dot{H}^{-1/4}(\mathbb R)$, so $\|(-\Delta)^{-1/8}f \|_{L^2}$ is an equivalent norm on $H^{-1/4}((0,1))$. Furthermore, the operator $(-\Delta)^{-1/8}$ is selfadjoint. Hence, $\left((-\Delta)^{-1/4}f,f\right) =\left((-\Delta)^{-1/8}f,(-\Delta)^{-1/8}f\right) = \left\|(-\Delta)^{-1/8}f\right\|_{L^2}^2\eqsim \|f\|_{H^{-1/4}((0,1))}^2$. $\endgroup$ – ifw Apr 14 '15 at 22:40
  • $\begingroup$ Thanks. Could you suggest a reference for (negative) fractional Laplacian and (negative) homogeneous Sobolev spaces? $\endgroup$ – F.M. Apr 16 '15 at 13:28
  • $\begingroup$ @F.M.: It depends on what you want to know about homogeneous Sobolev spaces, but you could possibly start with the first chapter of Fourier Analysis and Nonlinear Partial Differential Equations by Bahouri, Chemin, and Danchin. $\endgroup$ – ifw Apr 16 '15 at 23:34

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