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

摘要

In this paper we study Hardy and Rellich type inequalities forBaouendi-Grushin vector fields : $\nabla_{\gamma}=(\nabla_x,|x|^{2\gamma}\nabla_y)$ where $\gamma>0$, $\nabla_x$ and $\nabla_y$ are usualgradient 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 typeinequalities with remainder terms. In the second part, we prove two versions ofweighted Rellich type inequality on the whole space. We find sharp constantsfor these inequalities. We also obtain their improved versions for boundeddomains.