TY - JOUR

T1 - Geometric Hardy and Hardy-Sobolev inequalities on Heisenberg groups

AU - Ruzhansky, Michael

AU - Sabitbek, Bolys

AU - Suragan, Durvudkhan

N1 - Funding Information:
The first author was supported by the FWO Odysseus 1 Grant G.0H94.18N: Analysis and Partial Differential Equations and EPSRC Grant EP/R003025/1. The second author was supported in parts by the MES RK Grant AP08053051 and EPSRC Grant EP/R003025/1. The third author was supported by the Nazarbayev University program 091019CRP2120 and the Nazarbayev University Grant 240919FD3901. No new data was collected or generated during the course of this research.
Publisher Copyright:
© 2020 The Author(s).
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/12

Y1 - 2020/12

N2 - In this paper, we present geometric Hardy inequalities for the sub-Laplacian in half-spaces of stratified groups. As a consequence, we obtain the following geometric Hardy inequality in a half-space of the Heisenberg group with a sharp constant: Hu|pdζ ≥ p - 1 pp (ζ)p dist(ζ+)p|u|pdζ,p > 1, which solves a conjecture in the paper [S. Larson, Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domain in the Heisenberg group, Bull. Math. Sci. 6 (2016) 335-352]. Here, (ζ) =i=1n(X i(ζ),ν)2 + (Y i(ζ),ν)21 2 is the angle function. Also, we obtain a version of the Hardy-Sobolev inequality in a half-space of the Heisenberg group: -p - 1 pp (ζ)p dist(ζ)p|u|pdζ1 p ≥ C +|u|pdζ 1 p, where dist(ζ, +) is the Euclidean distance to the boundary, p:= Qp/(Q - p), and 2 ≤ p < Q. For p = 2, this gives the Hardy-Sobolev-Maz'ya inequality on the Heisenberg group.

AB - In this paper, we present geometric Hardy inequalities for the sub-Laplacian in half-spaces of stratified groups. As a consequence, we obtain the following geometric Hardy inequality in a half-space of the Heisenberg group with a sharp constant: Hu|pdζ ≥ p - 1 pp (ζ)p dist(ζ+)p|u|pdζ,p > 1, which solves a conjecture in the paper [S. Larson, Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domain in the Heisenberg group, Bull. Math. Sci. 6 (2016) 335-352]. Here, (ζ) =i=1n(X i(ζ),ν)2 + (Y i(ζ),ν)21 2 is the angle function. Also, we obtain a version of the Hardy-Sobolev inequality in a half-space of the Heisenberg group: -p - 1 pp (ζ)p dist(ζ)p|u|pdζ1 p ≥ C +|u|pdζ 1 p, where dist(ζ, +) is the Euclidean distance to the boundary, p:= Qp/(Q - p), and 2 ≤ p < Q. For p = 2, this gives the Hardy-Sobolev-Maz'ya inequality on the Heisenberg group.

KW - geometric Hardy inequality

KW - half-space

KW - Heisenberg group

KW - sharp constant

KW - Stratified groups

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U2 - 10.1142/S1664360720500162

DO - 10.1142/S1664360720500162

M3 - Article

AN - SCOPUS:85090528349

SN - 1664-3607

VL - 10

JO - Bulletin of Mathematical Sciences

JF - Bulletin of Mathematical Sciences

IS - 3

M1 - 2050016

ER -