I was hoping to find some references for the following facts:
Let $\Omega$ be a bounded domain with smooth boundary in $\mathbb{R}^n$.
(i) If $\nabla p$ is a distribution in $H^{-1}(\Omega)$ then $p \in L^2(\Omega)$.
(ii) Following two characterizations of $H^{-s}$ are equivalent: First is by taking the Fourier transform in distribution sense and consider the integral $\int (1+|\xi|^2)^{-s}|\hat{f}(\xi)|^2$, second is to take the dual of $H^s$ and consider the operator norm.
(iii) Spaces $H^{-s}$ are invariant under the action of a diffeomorphism on the space.