# Optimal Sobolev regularity for $(-\Delta)^{-s}$ on domains

In the paper Chen, Huyuan; Véron, Laurent, Semilinear fractional elliptic equations involving measures, (J. Differ. Equations 257, No. 5, 1457-1486 (2014). ZBL1290.35305) in the proof of Proposition 2.5 it is claimed that

For $$0 < \alpha < 1$$, the fractional operator $$(-\Delta)^{-\alpha}$$ is an isomorphism between $$W^{-\gamma,p}(\Omega)$$ and $$W^{2\alpha-\gamma}(\Omega)$$ for $$\gamma > N/p'$$ and $$p \in (1,\frac{N}{N+\beta-2\alpha})$$, where $$0 \leq \beta \leq \alpha$$.

The fractional Sobolev space is defined by the Gagliardo seminorm and duality for the negative order.

Is this correct, and how does one see this? Is there a reference? Moreover, is it true in general that $$(-\Delta)^{-s}$$ is continuous from $$H^{-s}(\mathbb{R}^N)$$ to $$H^{s}(\mathbb{R}^N)$$?

• The paper does indeed lack some explanations; I think you can make the quoted proof rigorous in some cases (conditions between differentiability and integrability) by incorporating a Dirichlet condition into $W^{2\alpha-\gamma,p}(\Omega)$ or adding the "$+1$". Since the result in question in the paper is (as far as I see it) never used again, it is a bit hard to tell where the authors wanted to go.. Jul 11, 2019 at 10:13

No. The operator $$(-∆)^s$$ is the Fourier multiplier $$\vert \xi\vert^{2s}$$ so that, say for $$f$$ in the Schwartz space whose Fourier transform vanishes near the origin, we have $$\Vert(-∆)^{-s} f\Vert_{H^s(\mathbb R^n)}^2=\int \vert\xi\vert^{-4s} \vert\hat f(\xi)\vert^2(1+\vert \xi\vert^2)^s,$$ $$\Vert f\Vert_{H^{-s}(\mathbb R^n)}^2=\int \vert\hat f(\xi)\vert^2(1+\vert \xi\vert^2)^{-s}.$$ As a result if $$\text{supp} \hat f\subset\{\xi, \epsilon\le \vert \xi\vert\le 2\epsilon\}$$, we have $$\Vert f\Vert_{H^{-s}(\mathbb R^n)}^2\approx\Vert f\Vert_{L^2(\mathbb R^n)}^2, \quad \Vert(-∆)^{-s} f\Vert_{H^s(\mathbb R^n)}^2\approx\epsilon^{-4s}\Vert f\Vert_{L^2(\mathbb R^n)}^2,$$ which are two inequivalent quantities (if $$s\not=0$$). On the other hand the operator $$(1-∆)^{-s}$$, which is the Fourier mutiplier $$(1+\vert \xi\vert^2)^{-s}$$ is obviously an isomorphism from $$H^{-s}(\mathbb R^n)$$ onto $$H^{s}(\mathbb R^n)$$.
• Thank you for your comment. It means that the argument in the paper ($(-\Delta)^s$ is an isomorphism from $W^{-\gamma,p}(\Omega)$ to $W^{2s-\gamma,p}(\Omega)$) is not correct, doesn’t it? Jul 10, 2019 at 16:18