The following inequality was proved firstly by Fefferman in the paper: The uncertainty principle, Bulletin of the AMS, 1983. Then it was improved by several authors. The proofs presented in these papers are quite involved. I wonder do we have a simple proof for such a nice inequality.
Let $\psi$ be a non-negative, measurable function on $\mathbb{R}^{n}$ satisfying $$ \left(\dfrac{1}{\left|B\right|}\intop_{B}\psi^{p}\right)^{\left.1\right/p}\leq C_{p}\left|B\right|^{-\left.2\right/n}, $$ for any open ball $B$ in $\mathbb{R}^{n}$. Here $p>1$ and $C_{p}>0$ are constants. We use $\left|B\right|$ for the volume of $B$. Then there is a positive constant $C$ such that $$ \intop_{\mathbb{R}^{n}}\left|f\right|^{2}\psi\leq C\intop_{\mathbb{R}^{n}}\left|\nabla f\right|^{2},\forall f\in C_{0}^{\infty}\left(\mathbb{R}^{n}\right). $$
