On extension problem, trace Hardy and Hardy's inequalities for some fractional Laplacians

Abstract

We obtain generalised trace Hardy inequalities for fractional powers of general operators given by sums of squares of vector fields. Such inequalities are derived by means of particular solutions of an extended equation associated to the above-mentioned operators. As a consequence, Hardy inequalities are also deduced. Particular cases include Laplacians on stratified groups, Euclidean motion groups and special Hermite operators. Fairly explicit expressions for the constants are provided. Moreover, we show several characterisations of the solutions of the extension problems associated to operators with discrete spectrum, namely Laplacians on compact Lie groups, Hermite and special Hermite operators.

An Extension Problem and Trace Hardy Inequality for the Sub-Laplacian on H-type Groups

“…Proof. The stated inequality for w ∈ C ∞ 0 (S) follows from the above proposition since −∆ S ≥ 1 4 (n + m) 2 . By approximating u and hence w by a sequence of C ∞ 0 (S) functions, we can conclude that the inequality remains true under the hypothesis on w. We have already remarked in Subsection 3.1 that when u satisfies the extension problem, then w(v, z, ρ) = ρ (n+m−s) 2…”

An extension problem and Hardy's inequality for the fractional Laplace-Beltrami operator on Riemannian symmetric spaces of noncompact type

“…Here also the constant is sharp and equality is achieved for the functions (δ 2 + |x| 2 ) −(n−2s)/2 and their translates [10].…”

“…Recently, in [37], Roncal and Thangavelu have proved analogues of Hardy-type inequalities with sharp constants for fractional powers of the sublaplacian on the Heisenberg group. For recent results on the Hardy-type inequalities for the fractional operators we refer [10,36,38].…”

An extension problem and Hardy's inequality for the fractional Laplace-Beltrami operator on Riemannian symmetric spaces of noncompact type

“…Here also the constant is sharp and equality is achieved for the functions (δ 2 + |x| 2 ) −(n−2s)/2 and their translates [10]. Generalization of the classical Hardy's inequality (1.2) to Riemannian manifolds was intensively pursued after the seminal work of Carron [12], see for instance [9,18,[29][30][31]43].…”