Conical square functionals on Riemannian manifolds

2021/01/06 by Thomas Cometx, Cometx, Thomas
Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2101.01922

openalex publication_date 2021/01/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let L = Δ+ V be Schrödinger operator with a non-negative potential V on a complete Riemannian manifold M. We prove that the conical square functional associated with L is bounded on L^p under different assumptions. This functional is defined by G_L (f) (x) = ( ∫₀^∞ ∫_B(x,t^1/2) |∇ e^-tL f(y)|² + V |e^-tL f(y)|² \fracdt dyVol(y,t^1/2) )^1/2.For p ∈ [2,+∞) we show that it is sufficient to assume that the manifold has the volume doubling property whereas for p ∈ (1,2) we need extra assumptions of L^p-L² of diagonal estimates for \ √(t) ∇ e^-tL, t≥ 0 \ and \ √(t) √(V) e^-tL , t ≥ 0\.Given a bounded holomorphic function F on some angular sector, we introduce the generalized conical vertical square functionalG_L^F (f) (x) = ( ∫₀^∞ ∫_B(x,t^1/2) |∇ F(tL) f(y)|² + V |F(tL) f(y)|² \fracdt dyVol(y,t^1/2) )^1/2 and prove its boundedness on L^p if F has sufficient decay at zero and infinity. We also consider conical square functions associated with the Poisson semigroup, lower bounds, and make a link with the Riesz transform.

Citations

• Hp spaces of several variables
• Littlewood-Paley-Stein functionals: an R-boundedness approach

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