Riesz transforms on solvable extensions of stratified groups
Volume 259 / 2021
Studia Mathematica 259 (2021), 175-200
MSC: 22E30, 42B20, 42B30.
DOI: 10.4064/sm190927-4-1
Published online: 26 April 2021
Abstract
Let $G = N \rtimes A$, where $N$ is a stratified group and $A = \mathbb {R}$ acts on $N$ via automorphic dilations. Homogeneous sub-Laplacians on $N$ and $A$ can be lifted to left-invariant operators on $G$ and their sum is a sub-Laplacian $\Delta $ on $G$. Here we prove weak type $(1,1)$, $L^p$-boundedness for $p \in (1,2]$ and $H^1 \to L^1$ boundedness of the Riesz transforms $Y \Delta ^{-1/2}$ and $Y \Delta ^{-1} Z$, where $Y$ and $Z$ are any horizontal left-invariant vector fields on $G$, as well as the corresponding dual boundedness results. At the crux of the argument are large-time bounds for spatial derivatives of the heat kernel, which are new when $\Delta $ is not elliptic.
