Let $u \in C^\infty_c(\mathbb{\Omega})$ and $\varphi$ be an eigenfunction of the fractional Laplacian $(-\Delta)^s$ in $\Omega$ with eigenvalue $\lambda$. In what sense, if any, is it true that $$\langle u, \varphi \rangle = \frac{1}{\lambda} \langle u, (-\Delta)^s\varphi \rangle = \frac{1}{\lambda} \langle(-\Delta)^s u, \varphi \rangle = \frac{1}{\lambda^2} \langle (-\Delta)^{2s} u, \varphi \rangle = ........?$$ In other words, is it true that we can integrate by parts using $(-\Delta)^su$, $(-\Delta)^{2s}u$ and so on even though their support is not compact anymore?

Let $u \in C^\infty_c(\mathbb{\Omega})$ and $\varphi$ be an eigenfunction of the fractional Laplacian $(-\Delta)^s$ in $\Omega$ with eigenvalue $\lambda$. In what sense, if any, is it true that $$\langle u, \varphi \rangle = \frac{1}{\lambda} \langle u, (-\Delta)^s\varphi \rangle = \frac{1}{\lambda} \langle(-\Delta)^s u, \varphi \rangle = \frac{1}{\lambda^2} \langle (-\Delta)^{2s} u, \varphi \rangle = ........?$$ Here $ \langle \cdot, \cdot \rangle$ denotes the scalar product in $L^2(\Omega)$. In other words, is it true that we can integrate by parts using $(-\Delta)^su$, $(-\Delta)^{2s}u$ and so on even though their support is not compact anymore?