From Modern Fourier Analysis by Grefakos, the homogeneous Sobolev space $\dot{L}_s^p(\mathbb R^n)$ is the space of all $u \in \mathcal S'(\mathbb R^n) / \mathscr P(\mathbb R^n)$ for which the well-defined distribution $\mathcal{F}^{-1}(|\xi|^s\widehat{u})$ conincides with a function in $\dot{L}^p(\mathbb R^n)$.
Here, $\mathscr P(\mathbb R^n)$ denotes a space of polynomials, $\dot{L}^p(\mathbb R^n)$ is the space of elements in $\mathcal S'(\mathbb R^n) / \mathscr P(\mathbb R^n)$ such that every equivalence class contains a unique representative that belongs to $L^p(\mathbb R^n)$.
I am confused by the expression $\mathcal{F}^{-1}(|\xi|^s\widehat{u})$, in particular, $|\xi|^s\widehat{u}$. By definition, we can multiply a $C^\infty$ function $h$ with $\partial^\alpha h$ having at most polynomial growth by a distribution $u$, which is defined as $\langle h u, \phi \rangle := \langle u, h\phi \rangle$. Since $h \in C^\infty$ with certain growth condition, we have $h\phi \in \mathcal S$.
However, $|\xi|^s$ can be problematic. For one thing, it may not even be defined at the origin, and even when it is defined, it may not be $C^\infty$. So, at the moment, I don't see how $|\xi|^s\widehat{u}$ can be a well-defined distribution.
I would appreciate if you could help me clear up my confusion.