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