The functional Loomis-Whitney type inequality in the Heisenberg groups and Projection theorems over finite fields

TL;DR

This work develops a functional Loomis–Whitney framework on finite Heisenberg groups $\mathbb{H}^n(\mathbb{F}_q)$, linking Loomis–Whitney–type inequalities to boundedness of multilinear projection forms and to orthogonal projection problems in a noncommutative setting. A sharp, complete exponent description is achieved for $n=1$, with the region $\frac{1}{u_1}+\frac{2}{u_2}\le 2$ and $\frac{2}{u_1}+\frac{1}{u_2}\le 2$, including the endpoint $L^{\tfrac{3}{2}}\times L^{\tfrac{3}{2}}\to L^1$, and a multilinear estimate at the critical exponent $u=\tfrac{n(2n+1)}{n+1}$ for general $n$ via fiberwise analysis and multilinear interpolation. Specializing to indicator functions yields a sharp Loomis–Whitney inequality bounding the size of a finite subset $K$ of $\mathbb{H}^n(\mathbb{F}_q)$ by the sizes of its $2n$ vertical projections, and via a straightening map this leads to covering statements by additive cosets and a novel approach to the orthogonal projection problem onto vertical hyperplanes. The results are optimal up to constants, with a planar $n=1$ refinement available through incidence bounds; the work also clarifies connections to boundedness problems for graphs with kernel operators and to noncommutative projection phenomena.

Abstract

We develop a functional Loomis--Whitney framework on the finite Heisenberg groups $\mathbb{H}^n(\mathbb{F}_q)$ and discover connections to the boundedness and orthogonal projection problems. For $n=1$ we determine the sharp region of exponents $(u_1,u_2)$ for which the associated bilinear projection form is bounded uniformly in $q$, namely \[ \frac{1}{u_1}+\frac{2}{u_2}\le 2 \quad\text{and}\quad \frac{2}{u_1}+\frac{1}{u_2}\le 2, \] which includes the endpoint $L^{3/2}\times L^{3/2}\to L^1$. For general $n$ we prove a multilinear estimate at the critical exponent \[ u=\frac{n(2n+1)}{n+1}, \] via an induction on $n$ that exploits the group's fiber structure together with multilinear interpolation. Specializing to indicators yields a sharp Loomis--Whitney type set inequality that controls $|K|$ for every finite $K\subset \mathbb{H}^n(\mathbb{F}_q)$ by the sizes of its $2n$ Heisenberg projections $\{π_j(K)\}$, forcing a large projection in every configuration. A straightening map then converts these bounds into covering statements by additive cosets and provides a new approach to the orthogonal projection problem onto the vertical hyperplanes $\{x_j=0\}$, which presents an interesting link between commutative and non-commutative settings. The obtained results are optimal up to absolute constants, and in the planar case $n=1$, when the size of the set is not too small, our bound can be sharpened further using a point--line incidence estimate.

Theorems & Definitions (28)

• Definition 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Corollary 1.5
• Theorem 1.6
• Definition 2.1
• Definition 2.2
• Lemma 2.3
• proof