Riesz transforms on $ax+b$ groups

Abstract

We prove the $L^p$-boundedness for all $p\in (1,\infty )$ of the first-order Riesz transforms $X_j{\mathcal{L}}^{-1/2}$ associated with the Laplacian $\mathcal{L}=-{\sum }_{j=0}^n{X}_j^2$ on the $ax+b$-group $G={\mathbb{R}}^n⋊\mathbb{R}$; here $X_0$ and $X_1,\dots ,X_n$ are left-invariant vector fields on $G$ in the directions of the factors $\mathbb{R}$ and ${\mathbb{R}}^n$ respectively. This settles a question left open in previous work of Hebisch and Steger (who proved the result for $p\le 2$) and of Gaudry and Sjögren (who only considered $n=1=j$). The main novelty here is that we can treat the case $p\in (2,\infty )$ and include the Riesz transform in the direction of $\mathbb{R}$; an operator-valued Fourier multiplier theorem on ${\mathbb{R}}^n$ turns out to be key to this purpose. We also establish a weak type $(1,1)$ endpoint for the adjoint Riesz transforms in the direction of ${\mathbb{R}}^n$. By transference, our results imply the $L^p$-boundedness for $p\in (1,\infty )$ of the first-order Riesz transforms associated with the Schrödinger operator $-{\partial }_s^2+e^{2s}$ on the real line.

40 pages

References in corpus (1)

• Riesz transform for a flow Laplacian on homogeneous trees