Loomis–Whitney inequalities in Heisenberg groups

Abstract This note concerns Loomis–Whitney inequalities in Heisenberg groups $$\mathbb {H}^n$$ Hn : $$\begin{aligned} |K| \lesssim \prod _{j=1}^{2n}|\pi _j(K)|^{\frac{n+1}{n(2n+1)}}, \qquad K \subset \mathbb {H}^n. \end{aligned}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">|K|≲∏j=12n|πj(K)|n+1n(2n+1),K⊂Hn. Here $$\pi _{j}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">πj , $$j=1,\ldots ,2n$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">j=1,…,2n are the vertical projections to hyperplanes $$\{x_j=0\}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">{xj=0} respectively, and $$|\cdot |$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">|·| refers a natural Haar measure on either or one of hyperplanes. The inequality first group {H}^1$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">H1 is direct consequence known $$L^p$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">Lp improving properties standard Radon transform {R}^2$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">R2 . In this note, we show how higher dimensional can be deduced by an elementary inductive argument from same approach, combined with multilinear interpolation, also yields following strong type bound: \int _{\mathbb {H}^n} _{j=1}^{2n} f_j(\pi _j(p))\;dp\lesssim \Vert f_j\Vert _{\frac{n(2n+1)}{n+1}} xmlns:mml="http://www.w3.org/1998/Math/MathML">∫Hn∏j=12nfj(πj(p))dp≲∏j=12n‖fj‖n(2n+1)n+1 for all nonnegative measurable functions $$f_1,\ldots ,f_{2n}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">f1,…,f2n {R}^{2n}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">R2n These their geometric corollaries thus ultimately based planar geometry. Among applications mention sharper version classical Sobolev u\Vert _{\frac{2n+2}{2n+1}} _{j=1}^{2n}\Vert X_ju\Vert ^{\frac{1}{2n}}, u \in BV(\mathbb {H}^n), xmlns:mml="http://www.w3.org/1998/Math/MathML">‖u‖2n+22n+1≲∏j=12n‖Xju‖12n,u∈BV(Hn), where $$X_j$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">Xj horizontal vector fields Finally, establish extension coordinate _1,\ldots ,\pi _{2n}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">π1,…,π2n replaced more general families mappings that allow us apply approach $$L^{3/2}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">L3/2 - $$L^3$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">L3 boundedness operator plane.