Let $(M,g)$ be a complete Riemannian manifold with Laplacian $\Delta:C^{\infty}_{c}(M)\to C^{\infty}_{c}(M)$ (think of $\mathbb{R}^{d}$ if you wish). This operator is essentially self-adjoint in $L^{2}(M)$ and hence, its closure $\overline{\Delta}:\mathcal{D}(\overline{\Delta})\to L^{2}(M)$ is self-adjoint. This allows us to define for any positive $s\in\mathbb{R}$ the operator $\overline{\Delta}^{s}$ by the spectral theorem of self-adjoint unbounded operators.

How to show that $C^{\infty}_{c}(M)$ is contained in the domain $\mathcal{D}(\overline{\Delta}^{s})$ for any $s$?

By the spectral theorem, the domain $\mathcal{D}(\overline{\Delta}^{s})$ is the set of all functions $f\in L^{2}(M)$ such that

$$\int_{\sigma(\overline{\Delta})}\lambda^{2s}\,\mathrm{d}\langle P(\lambda)f,f\rangle_{L^{2}}<\infty$$

where $P$ denotes the spectral measure of $\overline{\Delta}$. Now, for $f\in C^{\infty}_{c}(M)$, I don't see how to use its support of $f$ (information which is contained in the complex measure $\langle P(\lambda)f,f\rangle_{L^{2}}$) to show that the above integral is finite.