Weighted Rellich type inequalities related to Baouendi-Grushin operators

Abstract

In this paper we study Hardy and Rellich type inequalities for Baouendi-Grushin vector fields : ∇ γ = (∇ x , |x| 2γ ∇ y ) where γ > 0, ∇ x and ∇ y are usual gradient operators in the variables x ∈ R m and y ∈ R k , respectively. In the first part of the paper, we prove some weighted Hardy type inequalities with remainder terms. In the second part, we prove two versions of weighted Rellich type inequality on the whole space. We find sharp constants for these inequalities. We also obtain their improved versions for bounded domains.

Hardy inequalities for Landau Hamiltonian and for Baouendi-Grushin operator with Aharonov-Bohm type magnetic field. Part I

“…with sharp constant, which gives the result of D'Ambrosio (1.8) when p = 2 and Ω = R n . We also mention that inequality (2.15) has been established in [Kom15] and [SJ12].…”

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.…”

Weighted $$L^{p}$$ L p -Hardy and $$L^{p}$$ L p

“…Weighted Hardy and Rellich inequalities in different related contexts have been recently considered in [15] and [13]. For the general importance of such inequalities we can refer to [2].…”