Loomis-Whitney inequalities in Heisenberg groups

2021/04/14 by Fässler, Katrin, Pinamonti, Andrea · 1 citation
#28A75 #35R03 #46E35 #52C99 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2104.06684

Abstract

This note concerns Loomis-Whitney inequalities in Heisenberg groups ℍⁿ: |K| \lesssim ∏_j=1²ⁿ|π_j(K)|^{(n+1)/(n(2n+1))}, K ⊂ ℍⁿ. Here π_j, j=1,…,2n, are the vertical Heisenberg projections to the hyperplanes \x_j=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 L^p 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=1²ⁿ f_j(π_j(p)) dp\lesssim ∏_j=1²ⁿ ‖f_j‖_{(n(2n+1))/(n+1)} for all nonnegative measurable functions f₁,…,f_2n 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+2)/(2n+1)} \lesssim ∏_j=1²ⁿ‖X_ju‖^(1)/(2n), u ∈ BV(ℍⁿ), where X_j, j=1,…,2n, are the standard horizontal vector fields in ℍⁿ.

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• The functional Loomis-Whitney type inequality in the Heisenberg groups and Projection theorems over finite fields

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