The following is the definition of the space $H^s(\mathbb{R}^n)$ in Hunter's Applied Analysis:
Here a regular distribution is a tempered distribution $T_f$ such that it is given by $$ T_f(\varphi)=\int f(x)\varphi(x)\ dx,\quad \varphi\in \mathcal{S}(\mathbb{R}^n) $$ where $f$ is a Lebesgue measurable function on $\mathbb{R}^n$ such that $$ |f(x)|\leq g(x) (1+|x|^2)^{d/2} $$ a.e. for some nonegative integer $d\geq 0$ and nonnegative $g\in L^1$.
I have seen definition of $H^s$ that only the condition in the red box is assumed. For instance, see Tao's notes here.
Here is my question:
Are the two definitions (with or without the "regular distribution" assumption) of $H^s$ the same?
I think we are the same, I mainly studied this definition in "Real Analysis" by Folland. The definition can be justified as follows, note that the maps $\omega_s \cdot : \varphi(\xi) \in \mathcal{S}(\mathbb{R}^n) \longmapsto \omega_s(\xi)\varphi(\xi) \in \mathcal{S}(\mathbb{R}^n)$, where $\omega_s(\xi):=(1+|\xi|^2)^{s/2}$ is continuous with respect to convergence in Schwartz space, then we can define $\omega_s u \in \mathcal{S}'(\mathbb{R}^n)$ by $\langle \varphi , \omega_s u \rangle = \langle \omega_s \varphi, u \rangle$ $\forall \varphi \in \mathcal{S}(\mathbb{R}^n)$. The Fourier operator $\mathcal{F}: \mathcal{S}'(\mathbb{R}^n) \longrightarrow \mathcal{S}'(\mathbb{R}^n)$ is an isomorphism, extension of $\mathcal{F}: L^1(\mathbb{R}^n) \longrightarrow L^{\infty}(\mathbb{R}^n)$, and we can define $\Lambda^s : \mathcal{S}'(\mathbb{R}^n) \longrightarrow \mathcal{S}'(\mathbb{R}^n)$ by $\Lambda_s u = \mathcal{F}^{-1}(\omega_s \widehat{u})$ $\forall u \in \mathcal{S}'(\mathbb{R}^n)$, it is said Fourier multiplier with symbol $\omega_s$. The space $H^s(\mathbb{R}^n)$ is then defined as
$H^s(\mathbb{R}^n):=\lbrace u \in \mathcal{S}'(\mathbb{R}^n) : \Lambda^s u \in L^2(\mathbb{R}^n) \rbrace$
and by Plancherel theorem we have the norm
$\displaystyle \left \| u \right \|_{H^s}= \left \| \Lambda^s u \right \|_{L^2}= \left\| \mathcal{F}(\Lambda^s u) \right\|_{L^2}=\left \| \omega_s \widehat{u} \right \|_{L^2}=\left( \int_{\mathbb{R}^n} |\widehat{u}(\xi)|^2 (1+|\xi|^2)^s d\xi \right)^{1/2}$
I think that says "regular distributions" because in general we have continuous inclusion $L^{p}(\mathbb{R}^n) \hookrightarrow \mathcal{S}'(\mathbb{R}^n)$, with tempered distribution defined by "functions" $u_f$ or $T_f$ with your notation.