Riesz transforms on $ax+b$ groups

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 $G = \mathbb{R}^n \rtimes \mathbb{R}$; 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 $p \leq 2$) and of Gaudry and Sj\"ogren (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 Schr\"odinger operator $-\partial_s^2 + e^{2s}$ on the real line.