Planar incidences and geometric inequalities in the Heisenberg group

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

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