Hardy and Rellich type inequalities with remainders for Baouendi-Grushin vector fields

Abstract

In this paper we study Hardy and Rellich type inequalities for Baouendi-Grushin vector fields : ${\nabla }_{\gamma }=({\nabla }_x,|x{|}^{2\gamma }{\nabla }_y)$ where $\gamma >0$, ${\nabla }_x$ and ${\nabla }_y$ are usual gradient operators in the variables $x\in {\mathbb{R}}^m$ and $y\in {\mathbb{R}}^k$, 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.