Loomis–Whitney inequalities in Heisenberg groups

Abstract

AbstractThis note concernsLoomis–Whitney inequalitiesin Heisenberg groups$$\mathbb {H}^n$$Hn:$$\begin{aligned} |K| \lesssim \prod _{j=1}^{2n}|\pi _j(K)|^{\frac{n+1}{n(2n+1)}}, \qquad K \subset \mathbb {H}^n. \end{aligned}$$|K|≲∏j=12n|πj(K)|n+1n(2n+1),K⊂Hn.Here$$\pi _{j}$$πj,$$j=1,\ldots ,2n$$j=1,…,2n, are thevertical Heisenberg projectionsto the hyperplanes$$\{x_j=0\}$${xj=0}, respectively, and$$|\cdot |$$|·|refers to a natural Haar measure on either$$\mathbb {H}^n$$Hn, or one of the hyperplanes. The Loomis–Whitney inequality in the first Heisenberg group$$\mathbb {H}^1$$H1is a direct consequence of known$$L^p$$Lpimproving properties of the standard Radon transform in$$\mathbb {R}^2$$R2. 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$$\mathbb {H}^1$$H1. The same approach, combined with multilinear interpolation, also yields the following strong type bound:$$\begin{aligned} \int _{\mathbb {H}^n} \prod _{j=1}^{2n} f_j(\pi _j(p))\;dp\lesssim \prod _{j=1}^{2n} \Vert f_j\Vert _{\frac{n(2n+1)}{n+1}} \end{aligned}$$∫Hn∏j=12nfj(πj(p))dp≲∏j=12n‖fj‖n(2n+1)n+1for all nonnegative measurable functions$$f_1,\ldots ,f_{2n}$$f1,…,f2non$$\mathbb {R}^{2n}$$R2n. These inequalities and their geometric corollaries are thus ultimately based on planar geometry. Among the applications of Loomis–Whitney inequalities in$$\mathbb {H}^n$$Hn, we mention the following sharper version of the classical geometric Sobolev inequality in$$\mathbb {H}^n$$Hn:$$\begin{aligned} \Vert u\Vert _{\frac{2n+2}{2n+1}} \lesssim \prod _{j=1}^{2n}\Vert X_ju\Vert ^{\frac{1}{2n}}, \qquad u \in BV(\mathbb {H}^n), \end{aligned}$$‖u‖2n+22n+1≲∏j=12n‖Xju‖12n,u∈BV(Hn),where$$X_j$$Xj,$$j=1,\ldots ,2n$$j=1,…,2n, are the standard horizontal vector fields in$$\mathbb {H}^n$$Hn. Finally, we also establish an extension of the Loomis–Whitney inequality in$$\mathbb {H}^n$$Hn, where the Heisenberg vertical coordinate projections$$\pi _1,\ldots ,\pi _{2n}$$π1,…,π2nare replaced by more general families of mappings that allow us to apply the same inductive approach based on the$$L^{3/2}$$L3/2-$$L^3$$L3boundedness of an operator in the plane.