Let $f:\mathbb R^n \to \mathbb R$ be density function (i.e nonnegative function which integrates to $1$), and consider its Radon transform $R[f]$ defined by
$$ R[f](w,b) := \int_{\mathbb R^n}\delta(x^\top w - b)f(x)\,dx, $$
for any $(w,b) \in \mathbb R^{n+1}$.
Question. What are sufficient (and minimal) smoothness conditions on $f$ which ensure that for every $w \in \mathbb R$, the function $g_w:\mathbb R \to \mathbb R$ defined by $\overline{f}_w(b):= R[f](w,b)$ is continuously differentiable and $s(f):=\sup_{w,b} |\overline f'_w(b)| < \infty$. Also, what is a good upper-bound on $s(f)$ expressed in terms of some Sobolev norm of $f$ ?
Sanity check. The proposal should at least give miningful results for $f(x):= (2\pi\sigma^2)^{-n/2}e^{-\|x-c\|^2/(2\sigma^2)}$, the unnormalized Gaussian pdf on $\mathbb R^n$ with variance $\sigma^2I_n$ and mean $c \in \mathbb R^n$.
What is known
Thanks to this post https://mathoverflow.net/a/395470/78539, we know that if $f \in W^{k,p}(\mathbb R^n)$ for some $k \in \mathbb Z_+$ and $p \ge 1$ (finite!), then $$ \sup_w \|\overline f_w\|_{W^{k,p}(\mathbb R)} \le \|f\|_{W^{k,p}(\mathbb R^n)}. $$
However, this says nothing about $s(f)$, which would correspond to the case $(k,p)=(1,\infty)$.
