Abstract
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: [Formula: see text] 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, [Formula: see text] is the angle function. Also, we obtain a version of the Hardy–Sobolev inequality in a half-space of the Heisenberg group: [Formula: see text] where [Formula: see text] is the Euclidean distance to the boundary, [Formula: see text], and [Formula: see text]. For [Formula: see text], this gives the Hardy–Sobolev–Maz’ya inequality on the Heisenberg group.
Highlights
Let us recall the Hardy inequality in the half-space of Rn |∇u|pdx ≥ Rn+ p−1 p |u|p pRn+ xpn dx, p > 1, (1.1)for every function u ∈ C0∞(Rn+), where ∇ is the usual Euclidean gradient and Rn+ := {(x, xn)|x := (x1, . . . , xn−1) ∈ Rn−1, xn > 0}, n ∈ N
A Hardy inequality for a half-space of the Heisenberg group was shown by Luan and Young [11] in the form
An alternative proof of this inequality was given by Larson in [10], where the author generalized it to any half-space of the Heisenberg group
Summary
For every function u ∈ C0∞(Rn+), where ∇ is the usual Euclidean gradient and Rn+ := {(x , xn)|x := (x1, . . . , xn−1) ∈ Rn−1, xn > 0}, n ∈ N. A Hardy inequality for a half-space of the Heisenberg group was shown by Luan and Young [11] in the form. An alternative proof of this inequality was given by Larson in [10], where the author generalized it to any half-space of the Heisenberg group. Hardy inquality for the sub-Laplacian in the half-spaces of stratified groups (Carnot groups), where the obtained inequality will be a natural extension of the inequality derived by the authors in [10, 15] on Heisenberg and stratified groups, respectively. Geometric Lp-Hardy inequality on G+: Let G+ := {x ∈ G : x, ν > d} be a half-space of a stratified group G. Let us define the half-space on the stratified group G as G+ := {x ∈ G : x, ν > d}, where ν :=
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