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

2024/06/12 by Peter Sjögren, Sjögren, Peter, Maria Vallarino +1
Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2406.08032

openalex publication_date 2024/06/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be the Lie group ℝ²\rtimes 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=-∑_i=0²X_i² and the first-order Riesz transforms \mathcal R_i=X_iΔ^-1/2, \hskip3pt 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 \mathcal R_i. It is also proved that two of these Riesz transforms are bounded from H¹ to H¹.

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