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

Question

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 $H^{s}(\mathbb{R}^N)$?

Answer 1

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 $H^{s}(\mathbb R^n)$.