$L^p$ estimates for Baouendi–Grushin operators

Abstract

We prove $L^p$ estimates for the Baouendi–Grushin operator ${\mathrm{\Delta }}_x+|x{|}^{\alpha }{\mathrm{\Delta }}_y$ in $L^p\left({ℝ}^{N+M}\right)$, $1<p<\infty$, where $x\in {ℝ}^N$, $y\in {ℝ}^M$. When $p=2$ more general weights belonging to the reverse Hölder class $B_2\left({ℝ}^N\right)$ are allowed.