Hardy and Rellich inequalities for anisotropic p-sub-Laplacians

Abstract

In this paper we establish the subelliptic Picone type identities. As consequences, we obtain Hardy and Rellich type inequalities for anisotropic p-sub-Laplacians which are operators of the formwhere X i , i = 1, . . . , N , are the generators of the first stratum of a stratified (Lie) group. Moreover, analogues of Hardy type inequalities with multiple singularities and many-particle Hardy type inequalities are obtained on stratified groups.

Reverse integral Hardy inequality on metric measure spaces

“…In [28] and [29], the (direct) integral Hardy inequalities on metric measure space were established for 1 < p ≤ q < ∞ and 0 < q < p, 1 < p < ∞, respectively, with applications on homogeneous Lie groups, hyperbolic spaces, Cartan-Hadamard manifolds with negative curvature and on general Lie groups with Riemannian distance. Also, on Riemannian manifolds the Hardy inequality was obtained in [31], and on homogeneous Lie groups the Hardy inequality was obtained in [13], [24]- [23] and [30]. In the present paper, we continue the analysis in the general setting of metric measure spaces as in [28] and show the reverse integral Hardy inequality with q < 0 and p ∈ (0, 1).…”

“…By taking the second derivative of k(α) at the point α 1 and by denoting k 23) This means that, function k(α) has supremum at the point α = α 1 . Then, the biggest constant has the following relationship…”

Reverse Stein–Weiss, Hardy–Littlewood–Sobolev, Hardy, Sobolev and Caffarelli–Kohn–Nirenberg inequalities on homogeneous groups

“…The Hardy inequality was intensively studied. For example, the Hardy inequalities were considered on the Euclidean space in [19], [20], on the Heisenberg group in [6], [7], on stratified groups in [3], [31,32,33], [36], [42], for the vector fields in [34] and [38], on homogeneous groups in [37], [39]. In [37] (see e.g., [35]), authors showed the Hardy inequality with radial derivative on homogeneous groups in the following form: Theorem 1.4.…”

Finsler-Rellich inequalities involving the distance to the boundary

“…Concerning anisotropic (non-Riemannian) Rellich inequalities, there is a growing literature on inequalities with distance to a point, see e.g. [8,9,11] and references therein, but we are not aware of any results involving the distance to the boundary. To our knowlegde, the best Rellich constant for |∆u| p dx is not known even in the case of a half-space.…”