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

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

In the paper Semilinear fractional elliptic equations involving measures, 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 $H^{s}(\mathbb{R}^N)$?

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

I am reading a paper in fractional elliptic. It is true that $(-\Delta)^{-s}$ is continuous from $H^{-s}$ to $H^{s}$. However, in the following argument, I cannot find a good reference for it:

Fractional Paper

Can anyone explain or give me some references about the isomorphism?

Here is the full article: https://www.sciencedirect.com/science/article/pii/S002203961400196X

In the paper Semilinear fractional elliptic equations involving measures, 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 $H^{s}(\mathbb{R}^N)$?