Planar incidences and geometric inequalities in the Heisenberg group

Abstract

We prove that if $P,\mathcal{L}$ are finite sets of $\delta$-separated points and lines in ${\mathbb{R}}^2$, the number of $\delta$-incidences between $P$ and $\mathcal{L}$ is no larger than a constant times $$|P{|}^{2/3}|\mathcal{L}{|}^{2/3}\cdot {\delta }^{-1/3}.$$ We apply the bound to obtain the following variant of the Loomis-Whitney inequality in the Heisenberg group: $$|K|\lesssim |{\pi }_x(K){|}^{2/3}\cdot |{\pi }_y(K){|}^{2/3},K\subset \mathbb{H}.$$ Here ${\pi }_x$ and ${\pi }_y$ are the vertical projections to the $xt$- and $yt$-planes, respectively, and $|\cdot |$ refers to natural Haar measure on either $\mathbb{H}$, or one of the planes. Finally, as a corollary of the Loomis-Whitney inequality, we deduce that $$\Vert f{\Vert }_{4/3}\lesssim \sqrt{\Vert Xf\Vert \Vert Yf\Vert },f\in BV(\mathbb{H}),$$ where $X,Y$ are the standard horizontal vector fields in $\mathbb{H}$. This is a sharper version of the classical geometric Sobolev inequality $\Vert f{\Vert }_{4/3}\lesssim \Vert {\nabla }_{\mathbb{H}}f\Vert$ for $f\in BV(\mathbb{H})$.

18 pages