Loomis-Whitney inequalities in Heisenberg groups

Abstract

This note concerns Loomis–Whitney inequalities in Heisenberg groups Hn: |K|≲∏j=12n|πj(K)|n+1n(2n+1), K⊂Hn. Here πj, j=1,…,2n, are the vertical Heisenberg projections to the hyperplanes {xj=0}, respectively, and |⋅| refers to a natural Haar measure on either Hn, or one of the hyperplanes. The Loomis–Whitney inequality in the first Heisenberg group H1 is a direct consequence of known Lp improving properties of the standard Radon transform in 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 H1. The same approach, combined with multilinear interpolation, also yields the following strong type bound: ∫Hn∏j=12nfj(πj(p))dp≲∏j=12n‖fj‖n(2n+1)n+1 for all nonnegative measurable functions f1,…,f2n on R2n. These inequalities and their geometric corollaries are thus ultimately based on planar geometry. Among the applications of Loomis–Whitney inequalities in Hn, we mention the following sharper version of the classical geometric Sobolev inequality in Hn: ‖u‖2n+22n+1≲∏j=12n‖Xju‖12n,u∈BV(Hn), where Xj, j=1,…,2n, are the standard horizontal vector fields in Hn. Finally, we also establish an extension of the Loomis–Whitney inequality in Hn, where the Heisenberg vertical coordinate projections π1,…,π2n are replaced by more general families of mappings that allow us to apply the same inductive approach based on the L3/2-L3 boundedness of an operator in the plane.