Loomis-Whitney inequalities in Heisenberg groups

This note concerns Loomis-Whitney inequalities in Heisenberg groups $\mathbb{H}^n$: $$|K| \lesssim \prod_{j=1}^{2n}|\pi_j(K)|^{\frac{n+1}{n(2n+1)}}, \qquad K \subset \mathbb{H}^n.$$ Here $\pi_{j}$, $j=1,\ldots,2n$, are the vertical Heisenberg projections to the hyperplanes $\{x_j=0\}$, respectively, and $|\cdot|$ refers to a natural Haar measure on either $\mathbb{H}^n$, or one of the hyperplanes. The Loomis-Whitney inequality in the first Heisenberg group $\mathbb{H}^1$ is a direct consequence of known $L^p$ improving properties of the standard Radon transform in $\mathbb{R}^2$. In this note, we show how the Loomis-Whitney inequalities in higher dimensional Heisenberg groups can be deduced by an elementary inductive argument from the inequality in $\mathbb{H}^1$. The same approach, combined with multilinear interpolation, also yields the following strong type bound: $$\int_{\mathbb{H}^n} \prod_{j=1}^{2n} f_j(\pi_j(p))\;dp\lesssim \prod_{j=1}^{2n} \|f_j\|_{\frac{n(2n+1)}{n+1}}$$ for all nonnegative measurable functions $f_1,\ldots,f_{2n}$ on $\mathbb{R}^{2n}$. These inequalities and their geometric corollaries are thus ultimately based on planar geometry. Among the applications of Loomis-Whitney inequalities in $\mathbb{H}^n$, we mention the following sharper version of the classical geometric Sobolev inequality in $\mathbb{H}^n$: $$\|u\|_{\frac{2n+2}{2n+1}} \lesssim \prod_{j=1}^{2n}\|X_ju\|^{\frac{1}{2n}}, \qquad u \in BV(\mathbb{H}^n),$$ where $X_j$, $j=1,\ldots,2n$, are the standard horizontal vector fields in $\mathbb{H}^n$.