In Fefferman's "The Uncertainty Principle" paper (p.146), he mentions the "trivial" estimate $$ \int\limits_Q|\nabla u(x)|^2\,dx\geq\frac{c(\text{diam }Q)^{-2}}{|Q|}\int\limits_{Q\times Q}|u(x)-u(y)|^2\,dx\,dy\tag1 $$ where (in particular) $u\in C^1_0(\mathbb R^n)$ and $Q$ a cube in $\mathbb R^n$. However, I am having trouble understanding why the estimate is true. By Poincaré inequality on a ball, one easily obtains $$ \int\limits_B|\nabla u(x)|^2\,dx\geq\frac{c}{r^2}\frac{1}{|B|^2}\int\limits_B\left|\int\limits_B\big(u(x)-u(y))\,dy\,\right|^2\,dx\tag2 $$ where $B$ is a ball of radius $r$ (we note that it doesn't matter whether we take the estimate over balls or cubes...). But this seems strictly weaker than the aforementioned estimate, since the right-hand side of (2) is bounded above by the right-hand side of (1) by a trivial application of the Cauchy-Bunyakovski-Schwartz Inequality.
On the other hand, if (1) is true, then it is an improvement of Poincaré's inequality. It's not clear that such an improvement exists, but my knowledge is admittedly limited in this direction.
Any help proving (1)?