$L^p$-boundedness of Riesz transforms on solvable extensions of Carnot groups

TL;DR

This paper proves that Riesz transforms associated with sub-Laplacians on solvable extensions of Carnot groups are bounded on all L^p spaces for p in (1,∞), extending previous bounds from p in (1,2].

Contribution

The authors extend the L^p-boundedness of Riesz transforms to all p in (1,∞) on solvable extensions of Carnot groups using spectral multiplier theorems and Bessel function estimates.

Findings

Riesz transforms are bounded on L^p(G) for all p in (1,∞).

Extension of previous bounds from p in (1,2] to all p in (1,∞).

Application of operator-valued spectral multiplier theorem and Bessel function estimates.

Abstract

Let $G=N\rtimes \mathbb{R}$, where $N$ is a Carnot group and $\mathbb{R}$ acts on $N$ via automorphic dilations. Homogeneous left-invariant sub-Laplacians on $N$ and $\mathbb{R}$ can be lifted to $G$, and their sum is a left-invariant sub-Laplacian $\Delta$ on $G$. We prove that the first-order Riesz transforms $X \Delta^{-1/2}$ are bounded on $L^p(G)$ for all $p\in(1,\infty)$, where $X$ is any horizontal left-invariant vector field on $G$. This extends a previous result by Vallarino and the first-named author, who obtained the bound for $p\in(1,2]$. The proof makes use of an operator-valued spectral multiplier theorem, recently proved by the authors, and hinges on estimates for products of modified Bessel functions and their derivatives.