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 $\mathbb{R}<sup>{2}$, the number of $\delta$-incidences between $P$ and $\mathcal{L}$ is no larger than a constant times $$|P|^{{2/3}|\mathcal{L}|^{2/3}} \cdot \delta<sup>{-1/3}.$$ We apply the bound to obtain the following variant of the Loomis-Whitney inequality in the Heisenberg group: $$|K| \lesssim |\pi_{x}(K)|^{2/3} \cdot |\pi_{y}(K)|^{2/3}, \qquad K \subset \mathbb{H}.$$ 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 $$|f|<em>{4/3} \lesssim \sqrt{|Xf| |Yf| }, \qquad f \in BV(\mathbb{H}),$$ where $X,Y$ are the standard horizontal vector fields in $\mathbb{H}$. This is a sharper version of the classical geometric Sobolev inequality $|f|</em>{4/3} \lesssim |\nabla_{\mathbb{H}}f|$ for $f \in BV(\mathbb{H})$.