Riesz transforms on solvable extensions of stratified groups

TL;DR

This paper establishes boundedness properties of Riesz transforms on a class of solvable groups built from stratified groups, extending harmonic analysis tools to non-elliptic operators with new heat kernel estimates.

Contribution

It proves weak type (1,1), L^p-boundedness, and H^1 to L^1 boundedness of Riesz transforms on solvable extensions of stratified groups, including non-elliptic cases.

Findings

Proved weak type (1,1) and L^p-boundedness for p in (1,2]

Established H^1 to L^1 boundedness of Riesz transforms

Derived new large-time heat kernel derivative bounds for non-elliptic operators

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.