Grušin operators, Riesz transforms and nilpotent Lie groups

Abstract

We establish that the Riesz transforms of all orders corresponding to the Grušin operator $H_N=-\nabla _{x}^2-|x|^{2N}\,\nabla _{y}^2$, and the first-order operators $(\nabla _{x},x^\nu \,\nabla _{y})$ where $x\in \mathbf{R}^n$, $y\in \mathbf{R}^m$, $N\in \mathbf{N}_+$, and $\nu \in \{1,\ldots ,n\}^N$, are bounded on $L_p(\mathbf{R}^{n+m})$ for all $p\in \langle 1,\infty \rangle$ and are also weak-type (1, 1). Moreover, the transforms of order less than or equal to $N+1$ corresponding to $H_N$ and the operators $(\nabla _{x}, |x|^N\nabla _{y})$ are bounded on $L_p(\mathbf{R}^{n+m})$ for all $p\in \langle 1,\infty \rangle$. But if N is odd all transforms of order $N+2$ are bounded if and only if $p\in \langle 1,n\rangle$. The proofs are based on the observation that the $(\nabla _{x},x^\nu \,\nabla _{y})$ generate a finite-dimensional nilpotent Lie algebra, the corresponding connected, simply connected, nilpotent Lie group is isometrically represented on the spaces $L_p(\mathbf{R}^{n+m})$ and $H_N$ is the corresponding sublaplacian.