Given an integrable function (e.g a probability density function) $f:\mathbb R^n \to \mathbb R$, let $R[f]$ be its Radon transform defined by $$ R[f](w,b) := \int_{\mathbb R^n} \delta(x^\top w - b)f(x)\,dx, $$ for every $(w,b) \in \mathbb R^{n+1}$. Here, $\delta$ is the Dirac delta distribution.

Question. In terms of smoothness of $f$, what is a sufficient condition to ensure that (1) $\|R[f]\|_\infty < \infty$, and (2) there exist constants $\alpha,C \in (0,\infty)$ such that for every $w \in \mathbb R^d$, the function $b \mapsto R[f](w,b)$ is $(\alpha,C)$-Hölder continuous, i.e., $$ \big|R[f](w,b')-R[f](w,b)\big| \le C|b'-b|^\alpha, $$ for all $w \in \mathbb R^d$ and $b,b' \in \mathbb R$ ?

Related: Smoothness of Radon transform

Given an integrable function (e.g a probability density function) $f:\mathbb R^n \to \mathbb R$, let $R[f]$ be its Radon transform defined by $$ R[f](w,b) := \int_{\mathbb R^n} \delta(x^\top w - b)f(x)\,dx, $$ for every $(w,b) \in \mathbb R^{n+1}$. Here, $\delta$ is the Dirac delta distribution.

Question. In terms of smoothness of $f$, what is a sufficient condition to ensure that (1) $\|R[f]\|_\infty < \infty$, and (2) there exist constants $\alpha,C \in (0,\infty)$ such that for every $w \in \mathbb R^d$, the function $b \mapsto R[f](w,b)$ is $(\alpha,C)$-Hölder continuous, i.e., $$ \big|R[f](w,b')-R[f](w,b)\big| \le C|b'-b|^\alpha, \tag{+} $$ for all $w \in \mathbb R^d$ and $b,b' \in \mathbb R$ ?

Related: Smoothness of Radon transform