Riesz transforms on non-compact manifolds

2014/11/01 by Chen Peng, Chen, Peng, Jocelyn Magniez +3
Mathematics · #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1411.0137

openalex publication_date 2014/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let M be a complete non-compact Riemannian manifold satisfying the doubling volume property as well as a Gaussian upper bound for the corresponding heat kernel. We study the boundedness of the Riesz transform dΔ^-(1)/(2) on both Hardy spaces H^p and Lebesgue spaces L^p under two different conditions on the negative part of the Ricci curvature R^-. First we prove that if R^- is α-subcritical for some α∈ [0,1), then the Riesz transform d^*Δ^-(1)/(2) on differential 1-forms is bounded from the associated Hardy space H^p_{\overrightarrowΔ}(Λ¹T^*M) to L^p(M) for all p∈ [1,2]. As a consequence, the Riesz transform (on functions) is bounded on L^p for all p∈ (1,p₀) where p₀>2 depends on α and the constant appearing in the doubling property. Second, we prove that if ∫₀¹ ‖\frac|R^-|^(1)/(2)v(⋅, √(t))^(1)/(p₁)‖_p1(dt)/(√(t))+∫₁^∞ ‖\frac|R^-|^(1)/(2)v(⋅, √(t))^(1)/(p₂)‖_p2(dt)/(√(t))2 and p₂>3, then the Riesz transform dΔ^-(1)/(2) is bounded on L^p for all 1 0, then dΔ^-(1)/(2) is bounded on L^p for all 12 under conditions on R^- and the potential V. We prove both positive and negative results on the boundedness of dA^-(1)/(2) on L^p

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