I am working on a problem where I have continuous probability distributions $p$ over a bounded domain $D$, i.e., $\forall x$ $p(x) \geq 0$ and $\|p\|_1 = \int_D p(x) dx = 1$. However, I also want $p$ to have bounded function 2-norm, i.e., $\|p\|_2 = \sqrt{\int_D p(x)^2 dx} \leq c$ for some finite $c$. Is there a name for such a class of distributions? Edit: Assuming $p \in L^2$, can one bound $\|p\|_2$ with some other quantities involving $p$? I provide some examples in the following paragraph.
The trivial bound is $\|p\|_2 \leq \|p\|_\infty {\rm Vol}(D)$. I am looking for something better, if possible. It seems that exponential distributions with large-enough spread have low 2-norm. Also, for uniform distributions one can easily get $\|p\|_2$ as a function of the volume of the domain $D$. I am looking for any resources / references along this direction. Please help.
Edit: As pointed out in the comments, I looked at the Plancherel Identity, and realize that $p \in L^2$ iff the characteristic function of the random variable $X$ having density $p$ is in $L^2$. However, I am looking for a simpler bound on $\|p\|_2$, or a condition on $p$ that is simpler than assuming the characteristic function is square integrable.
