Gru\v{s}in operators, Riesz transforms and nilpotent Lie groups

We establish that the Riesz transforms of all orders corresponding to the Gru\v{s}in operator $H_N=-\nabla_{x}^2-|x|^{2N}\,\nabla_{y}^2$, and the first-order operators $(\nabla_{x},x^\nu\,\nabla_{y})$ where $x\in \Ri^n$, $y\in\Ri^m$, $N\in\Ni_+$, and $\nu\in\{1,\ldots,n\}^N$, are bounded on $L_p(\Ri^{n+m})$ for all $p\in\langle1,\infty\rangle$ 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 $L_p(\Ri^{n+m})$ for all $p\in\langle1,\infty\rangle$. But all transforms of order $N+2$ are bounded if and only if $p\in\langle1,n\rangle$. 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 $L_p(\Ri^{n+m})$ and $H_N$ is the corresponding sublaplacian