Hardy spaces associated with Schrodinger operators on the Heisenberg group

2011/06/24 by Chin-Cheng Lin, Lin, Chin-Cheng, Heping Liu +3
Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Analysis and Transform Methods

paper · pdf · doi:10.48550/arxiv.1106.4960

openalex publication_date 2011/06/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let L= -Δ_ℍⁿ+V be a Schrödinger operator on the Heisenberg group ℍⁿ, where Δ_ℍⁿ is the sub-Laplacian and the nonnegative potential V belongs to the reverse Hölder class B_(Q)/(2) and Q is the homogeneous dimension of ℍⁿ. The Riesz transforms associated with the Schrödinger operator L are bounded from L¹(ℍⁿ) to L^1,∞(ℍⁿ). The L¹ integrability of the Riesz transforms associated with L characterizes a certain Hardy type space denoted by H¹_L(ℍⁿ) which is larger than the usual Hardy space H¹(ℍⁿ). We define H¹_L(ℍⁿ) in terms of the maximal function with respect to the semigroup \e^{-s L}: s>0 \, and give the atomic decomposition of H¹_L(ℍⁿ). As an application of the atomic decomposition theorem, we prove that H¹_L(ℍⁿ) can be characterized by the Riesz transforms associated with L. All results hold for stratified groups as well.

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