Let $f:\mathbb R^n \to \mathbb R$ be a probability density function, meaning that $f \ge 0$ and $f \in L^1(\mathbb R^n)$, and define its Radon transform $R[f]$ by $$ R[f](w,b) := \int_{\mathbb R^n}\delta(x^\top w - b)f(x)\,dx, $$ for every $(w,b) \in \mathbb R^{n+1}$ with $\|w\| = 1$.
Question. What is a necessary and sufficient condition on $f$ which ensures that $\|R[f]\|_\infty:=\sup_{w,b} R[f](w,b) < \infty$ ?
Known results
- $\|R[f]\|_{\infty,1} := \sup_w \|R[f](w,\cdot)\|_1 \le \|f\|_1 < \infty$.