Let $\gamma \geq 0$ and consider the fractional derivative operator defined in Fourier domain by $$\mathcal{F} \{\mathrm{D}^{\gamma} \varphi \} (\omega) = (\mathrm{i} \omega)^{\gamma} \mathcal{F}\{\varphi\} (\omega),$$ where $\varphi \in \mathcal{S}(\mathbb{R})$ is a smooth and rapidly decaying function.

Of course, the definition can be extended to much more functions than $\varphi \in \mathcal{S}(\mathbb{R})$, including some, but not all, tempered distributions. It is for instance possible to extend $\mathrm{D}^{\gamma}$ to any compactly supported distribution (as for any convolution operator from $\mathcal{S}(\mathbb{R})$ to $\mathcal{S}'(\mathbb{R})$).

My question is the following: Is there a good notion of the "domain of definition" of the operator $\mathrm{D}^{\gamma}$, understood as the largest topological vector space $\mathcal{S}(\mathbb{R}) \subseteq \mathcal{X} \subseteq \mathcal{S}'(\mathbb{R})$ such that $\mathrm{D}^{\gamma} : \mathcal{X} \rightarrow \mathcal{S}'(\mathbb{R})$ is well-defined and continuous? Or at least, if the question is somehow meaningless, any natural construction that will include many tempered distributions in a satisfactory* manner?

*To give a bit of context, I am especially interested by the fractional case where $\gamma \notin \mathbb{N}$. The question is obvious for $\gamma = n \in \mathbb{N}$, since one can select $\mathcal{X} = \mathcal{S}'(\mathbb{R})$. However. when $\gamma$ is purely fractional, there is no hope to define the product $(\mathrm{i} \omega)^{\gamma} \mathcal{F}\{u\} (\omega)$ when $u \in \mathcal{S}'(\mathbb{R})$ is too irregular around the origin, which means morally that $u$ growth too fast at infinity. "In a satisfactory manner" would be a way of specifying properly a good "growth property" of $u \in \mathcal{X}$.