Conditions that ensure a bounded probability density function

Question

Let $X$ be a random variable that is absolutely continuous with respect to the Lebesgue measure. That means that $X$ has a PDF $f$. Are there simple conditions on $X$ that ensure that $$ \sup_{x\in\mathbb{R}}f(x) < \infty. $$ with simple conditions. I mean for example, certain moment conditions or other tractable assumptions.

Answer 1

Let the CDF be Lipschitz continuous.

If $|F(x)-F(y)|\leq K|x-y|$ for every $x,y\in\mathbb R$ and $g$ serves as PDF for the distribution then also the function $f$ prescribed by $x\mapsto\min(K,g(x))$ will serve as PDF of the distribution.

This with: $$\sup_{x\in\mathbb R}f(x)\leq K<\infty$$