Riesz transforms on solvable extensions of stratified groups

Abstract

Let $G=N⋊A$, 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.

20 pages. This work sharpens and extends heat kernel estimates initially proved in arXiv:1504.03862

References in corpus (3)

• Spectral multipliers for sub-Laplacians on solvable extensions of stratified groups
• Analysis on harmonic extensions of H-type groups
• Spaces H^1 and BMO on ax+b-groups

Cited by in corpus (3)

• Riesz transform for a flow Laplacian on homogeneous trees
• Riesz transforms on $ax+b$ groups
• Hardy spaces and Riesz transforms on a Lie group of exponential growth