The standard way to define the homogeneous Sobolev space $\dot{H}^m(\mathbb R^n)$ for $m\in \mathbb N$ is via the condition $f\in \dot{H}^m(\mathbb R^n)$ if and only if $f \in \mathscr S' (\mathbb R^n)$ (the Schwartz space) and $\||\xi|^m \hat f\|_{L^2} < \infty$. Recently, I came across the following definition
$$ \dot{H}^m(\mathbb R^n) = \{f\in \mathscr D' (\mathbb R^n)| \nabla^m f\in L^2(\mathbb R^n)\}, $$ where $\mathscr D' (\mathbb R^n)$ denotes the space of distributions.
Is it possible to show that those two definitions are equivalent? One direction is trivial since $\mathscr D (\mathbb R^n)\subset \mathscr S (\mathbb R^n)$ and so $\mathscr S' (\mathbb R^n)\subset \mathscr D' (\mathbb R^n)$.
Edit We also know $\dot H ^m (\mathbb R^n) \subset L^2_{loc} (\mathbb R^n)$. This follows from the fact that a distribution with partial derivatives in $L^2_{loc}$ is in $L^2_{loc}$.
Thanks for any hints :)
Fourier-Plancherel formula says that $\|\hat{f}\|_{L^{2}}=\|f\|_{L^{2}}$, where $\hat{f}(\xi)=\int_{\mathbb{R}^{d}}f(x)e^{-2\pi ix\cdot\xi}dx$. In addition, we also have $\widehat{\partial^{\alpha}f}=(2\pi i\xi)^{\alpha}\hat{f}(\xi)$. So combining the two above facts, you can get the equivalence of two definitions.