It turns out that the "correct" domain of definition of the Radon transform is the Schwarz space $\mathcal S(\mathbb R^n)$ of infinitely-differential functions on $\mathbb R^n$ with derivatives which decrease faster than any polynomial.

Also, it is well-known that for any $\mathbb R$, $\mathcal S(\mathbb R^n)$ is dense in $H^k(\mathbb R^d)$, the fractional Sobolev space of order $k$. Taking $\alpha \in (0,1)$, we know thanks to Lemma 1 of Oberlin and Stein (1982) Mapping Properties of the Radon Transform that, there exists a constant $C_\alpha \in (0,\infty)$ such that for any appropriately integrable function $g:\mathbb R \to \mathbb R$, it holds that $$ \sup_{b \in \mathbb R}|\Delta_t g(b)| \le C_\alpha\|g\|_{H^k(\mathbb R)}\cdot |t|^\alpha, \tag{1} $$ where $k := \alpha+1/2$ and $\Delta_t g(b) := g(b+t)-g(b)$. Applying (1) with $g = R_w[f]:b \mapsto R[f](w,b)$, for fixed nonzero $w \in \mathbb R^n$, and recalling the Central Slice Theorem by which $\widehat{R_w[f]}(s) = \|w\|_2\cdot \widehat{f}(sw)$, we deduce that $$ \frac{|\Delta_t R_w[f](b)|^2}{C_\alpha^2|t|^{2\alpha}} \le \ldots \le \|w\|^{-2\alpha}\|f\|_{H^k(\mathbb R^n)}, $$ from which it follows that $R_w[f]$ is Hoelder-continuous of order $\alpha$.

It turns out that the "correct" domain of definition of the Radon transform is the Schwarz space $\mathcal S(\mathbb R^n)$ of infinitely-differential functions on $\mathbb R^n$ with derivatives which decrease faster than any polynomial.

Also, it is well-known that for any integer $k$, $\mathcal S(\mathbb R^n)$ is dense in $H^k(\mathbb R^n)$, the fractional Sobolev space of order $k$. Taking $\alpha \in (0,1)$, we know thanks to Lemma 1 of Oberlin and Stein (1982) Mapping Properties of the Radon Transform that, there exists a constant $C_\alpha \in (0,\infty)$ such that for any appropriately integrable function $g:\mathbb R \to \mathbb R$, it holds that $$ \sup_{b \in \mathbb R}|\Delta_t g(b)| \le C_\alpha\|g\|_{H^k(\mathbb R)}\cdot |t|^\alpha, \tag{1} $$ where $k := \alpha+1/2$ and $\Delta_t g(b) := g(b+t)-g(b)$. Applying (1) with $g = R_w[f]:b \mapsto R[f](w,b)$, for fixed nonzero $w \in \mathbb R^n$, and recalling the Central Slice Theorem by which $\widehat{R_w[f]}(s) = \|w\|_2\cdot \widehat{f}(sw)$, we deduce that $$ \frac{|\Delta_t R_w[f](b)|^2}{C_\alpha^2|t|^{2\alpha}} \le \ldots \le \|w\|^{-2\alpha}\|f\|_{H^k(\mathbb R^n)}, $$ from which it follows that $R_w[f]$ is Hoelder-continuous of order $\alpha$.