Grušin operators, Riesz transforms and nilpotent Lie groups

Abstract

We establish that the Riesz transforms of all orders corresponding to the GruÅ¡in operator $H_N=-{\nabla }_x^2-|x{|}^{2N}{\nabla }_y^2$, and the first-order operators $({\nabla }_x,x^{\hat{I}}½{\nabla }_y)$ where $x\in \Ri^n$, $y\in\Ri^m$, $N\in\Ni_+$, and $\hat{I}½\in \{1,\dots ,n{\}}^N$, are bounded on $L_p(\Ri^{n+m})$ for all $p\in ⟨1,\infty ⟩$ 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 $p\in ⟨1,\infty ⟩$. But all transforms of order $N+2$ are bounded if and only if $p\in ⟨1,n⟩$. The proofs are based on the observation that the $({\nabla }_x,x^{\hat{I}}½{\nabla }_y)$ 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

11 pages