Loomis-Whitney inequalities in Heisenberg groups

Abstract: This note concerns Loomis-Whitney inequalities in Heisenberg groups $\mathbb{H}<sup>n$: $$|K| \lesssim \prod_{j=1}^{{2n}|\pi_j(K)|^{{\frac{n+1}{n(2n+1)}},}} \qquad K \subset \mathbb{H}<sup>n.$$ 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 $\mathbb{H}<sup>n$, or one of the hyperplanes. The Loomis-Whitney inequality in the first Heisenberg group $\mathbb{H}<sup>1$ is a direct consequence of known $L<sup>p$ improving properties of the standard Radon transform in $\mathbb{R}<sup>2$. 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 $\mathbb{H}<sup>1$. The same approach, combined with multilinear interpolation, also yields the following strong type bound: $$\int_{\mathbb{H}^n} \prod_{j=1}^{2n} f_j(\pi_j(p))\;dp\lesssim \prod_{j=1}^{2n} |f_j|<em>{\frac{n(2n+1)}{n+1}}$$ for all nonnegative measurable functions $f_1,\ldots,f</em>{2n}$ on $\mathbb{R}<sup>{2n}$. These inequalities and their geometric corollaries are thus ultimately based on planar geometry. Among the applications of Loomis-Whitney inequalities in $\mathbb{H}<sup>n$, we mention the following sharper version of the classical geometric Sobolev inequality in $\mathbb{H}<sup>n$: $$|u|<em>{\frac{2n+2}{2n+1}} \lesssim \prod</em>{j=1}^{{2n}|X_ju|^{{\frac{1}{2n}},}} \qquad u \in BV(\mathbb{H}<sup>n),$$ where $X_j$, $j=1,\ldots,2n$, are the standard horizontal vector fields in $\mathbb{H}<sup>n$.