Planar incidences and geometric inequalities in the Heisenberg group

TL;DR

This paper establishes new incidence bounds for points and lines in the plane, applies them to derive a Loomis-Whitney inequality in the Heisenberg group, and improves geometric Sobolev inequalities for functions on this group.

Contribution

It introduces a novel incidence bound in the Euclidean plane and uses it to derive geometric inequalities in the Heisenberg group, connecting combinatorial geometry with sub-Riemannian analysis.

Findings

Incidence bound: $|P|^{2/3}| ext{lines}|^{2/3} imes ext{scale}^{-1/3}$

Loomis-Whitney inequality in $ ext{Heisenberg group}$: $|K| ot hickapprox | ext{projections}|^{2/3}$

Sharper Sobolev inequality: $ orm{f}_{4/3} ot hickapprox orm{Xf}^{1/2} orm{Yf}^{1/2}$

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: $$ |K| \lesssim |\pi_{x}(K)|^{2/3} \cdot |\pi_{y}(K)|^{2/3}, \qquad K \subset \mathbb{H}. $$ 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 $$ \|f\|_{4/3} \lesssim \sqrt{\|Xf\| \|Yf\| }, \qquad f \in BV(\mathbb{H}), $$ 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 $f \in BV(\mathbb{H})$.