The Two Weight Inequality for Poisson Semigroup on Manifold with Ends

2021/03/03 by Xuan Thinh Duong, Duong, Xuan Thinh, Ming-Yi Lee +5
Computer Science · Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2103.02292

openalex publication_date 2021/03/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let M = \mathbb R^m \sharp \mathcal Rⁿ be a non-doubling manifold with two ends \mathbb R^m \sharp \mathcal Rⁿ, m > n ≥ 3. Let Δ be the Laplace--Beltrami operator which is non-negative self-adjoint on L²(M). Then Δ and its square root √Δ generate the semigroups e^-tΔ and e^-t√Δ on L²(M), respectively. We give testing conditions for the two weight inequality for the Poisson semigroup e^-t√Δ to hold in this setting. In particular, we prove that for a measure μ on M₊:=M× (0,∞) and σ on M: ‖P_σ(f)‖_L²(M₊;μ) \lesssim ‖f‖_L²(M;σ), with P_σ(f)(x,t):= ∫_M P_t(x,y)f(y) dσ(y) if and only if testing conditions hold for the Poisson semigroup and its adjoint. Further, the norm of the operator is shown to be equivalent to the best constant in these testing conditions.

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