Hardy spaces and Riesz transforms on a Lie group of exponential growth

Journal article, 2025

Let G be the Lie group R2⋊R+R2⋊R+ endowed with the Riemannian symmetric space structure. Take a distinguished basis X0,X1,X2 of left-invariant vector fields of the Lie algebra of G, and consider the Laplacian Δ=−∑2i=0X2i and the first-order Riesz transforms Ri=XiΔ−1/2, i=0,1,2. We first show that the atomic Hardy space H1 in G introduced by the authors in a previous paper does not admit a characterization in terms of the Riesz transforms RiRi. It is also proved that two of these Riesz transforms are bounded from H1 to H1.