Boundedness from H^1 to L^1 of Riesz transforms on a Lie group of exponential growth

TL;DR

This paper investigates the boundedness properties of first and second order Riesz transforms on a specific Lie group with exponential growth, establishing which operators map the Hardy space H^1 to L^1.

Contribution

It provides new results on the boundedness of Riesz transforms on a Lie group of exponential growth, distinguishing between operators that are bounded from H^1 to L^1 and those that are not.

Findings

Operators R_i are bounded from H^1 to L^1.

Operators S_i are not bounded from H^1 to L^1.

Second order transforms T_{ij} are bounded from H^1 to L^1.

Abstract

Let $G$ be the Lie group given by the semidirect product of $R^2$ and $R^+$ endowed with the Riemannian symmetric space structure. Let $X_0, X_1, X_2$ be a distinguished basis of left-invariant vector fields of the Lie algebra of $G$ and define the Laplacian $\Delta=-(X_0^2+X_1^2+X_2^2)$. In this paper we consider the first order Riesz transforms $R_i=X_i\Delta^{-1/2}$ and $S_i=\Delta^{-1/2}X_i$, for $i=0,1,2$. We prove that the operators $R_i$, but not the $S_i$, are bounded from the Hardy space $H^1$ to $L^1$. We also show that the second order Riesz transforms $T_{ij}=X_i\Delta^{-1}X_j$ are bounded from $H^1$ to $L^1$, while the Riesz transforms $S_{ij}=\Delta^{-1}X_iX_j$ and $R_{ij}=X_iX_j\Delta^{-1}$ are not.