Loomis-Whitney inequalities in Heisenberg groups
Fässler, K., & Pinamonti, A. (2022). Loomis-Whitney inequalities in Heisenberg groups. Mathematische Zeitschrift, 301(2), 1983-2010. https://doi.org/10.1007/s00209-022-02968-y
Published in
Mathematische ZeitschriftDate
2022Copyright
© The Author(s) 2022
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.
...


Publisher
Springer Science and Business Media LLCISSN Search the Publication Forum
0025-5874Keywords
Publication in research information system
https://converis.jyu.fi/converis/portal/detail/Publication/104220297
Metadata
Show full item recordCollections
Related funder(s)
Academy of FinlandFunding program(s)
Research costs of Academy Research Fellow, AoF; Research post as Academy Research Fellow, AoF
Additional information about funding
K. Fässler is supported by the Academy of Finland via the project Singular integrals, harmonic functions, and boundary regularity in Heisenberg groups, grant Nos. 321696, 328846. A. Pinamonti is partially supported by supported by the University of Trento and GNAMPA of INDAM.License
Related items
Showing items with similar title or keywords.
-
Extensions and corona decompositions of low-dimensional intrinsic Lipschitz graphs in Heisenberg groups
Di Donato, Daniela; Fässler, Katrin (Springer, 2022)This note concerns low-dimensional intrinsic Lipschitz graphs, in the sense of Franchi, Serapioni, and Serra Cassano, in the Heisenberg group Hn, n∈N. For 1⩽k⩽n, we show that every intrinsic L-Lipschitz graph over a subset ... -
Assouad Type Dimensions in Geometric Analysis
Lehrbäck, Juha (Birkhäuser, 2021)We consider applications of the dual pair of the (upper) Assouad dimension and the lower (Assouad) dimension in analysis. We relate these notions to other dimensional conditions such as a Hausdorff content density condition ... -
Plenty of big projections imply big pieces of Lipschitz graphs
Orponen, Tuomas (Springer, 2021)I prove that closed n-regular sets E⊂Rd with plenty of big projections have big pieces of Lipschitz graphs. In particular, these sets are uniformly n-rectifiable. This answers a question of David and Semmes from 1993. -
Planar Sobolev extension domains
Zhang, Yi (University of Jyväskylä, 2017)This doctoral thesis deals with geometric characterizations of bounded planar simply connected Sobolev extension domains. It consists of three papers. In the first and third papers we give full geometric characterizations ... -
A density result on Orlicz-Sobolev spaces in the plane
Ortiz, Walter A.; Rajala, Tapio (Elsevier, 2021)We show the density of smooth Sobolev functions Wk,∞(Ω)∩C∞(Ω) in the Orlicz-Sobolev spaces Lk,Ψ(Ω) for bounded simply connected planar domains Ω and doubling Young functions Ψ.