Subelliptic geometric Hardy type inequalities on half-spaces and convex domains

Michael Ruzhansky,Bolys Sabitbek,Durvudkhan Suragan

doi:10.1007/s43034-020-00067-9

Abstract

In this paper we present L^2 and L^p versions of the geometric Hardy inequalities in half-spaces and convex domains on stratified (Lie) groups. As a consequence, we obtain the geometric uncertainty principles. We give examples of the obtained results for the Heisenberg and the Engel groups.

Highlights

In the Euclidean setting, a geometric Hardy inequality in a (Euclidean) convex domain Ω has the following formCommunicated by Joachim Toft.Almaty, Kazakhstan 4 Department of Mathematics, School of Sciences and Humanities, Nazarbayev University, 53Kabanbay Batyr Ave, 010000 Nursultan, KazakhstanVol:.(1234567890)Subelliptic geometric Hardy type inequalities
In this note by using the approach in [11] we obtain the geometric Hardy type inequalities on the half-spaces and the convex domains on general stratified groups, so our results extend known results of Abelian (Euclidean) and Heisenberg groups
Let G = (Rn, ◦, ) be a stratified Lie group, with dilation structure and Jacobian generators X1, ... , XN, so that N is the dimension of the first stratum of

Summary

Department of Mathematics

In the case of the Heisenberg group H , Luan and Yang [12] obtained the following Hardy inequality on the half space H+ ∶= {(x1, x2, x3) ∈ H, | x3 > 0} for u ∈ C0∞(H+). The geometric Lp-Hardy inequalities for the sub-Laplacian on the convex domain in the Heisenberg group was obtained by Larson [11] which generalises the previous result in [12]. In this note by using the approach in [11] we obtain the geometric Hardy type inequalities on the half-spaces and the convex domains on general stratified groups, so our results extend known results of Abelian (Euclidean) and Heisenberg groups. 2 we present L2 and Lp versions of the subelliptic geometric Hardy type inequalities on the half-space. 3, we show subelliptic L2 and Lp versions of the geometric Hardy type inequalities on the convex domains

Preliminaries

L2‐Hardy inequality on the half‐space of

Hardy inequalities on a convex domain of

Lp‐Hardy’s inequality on a convex domain of