Riesz transform and spectral multipliers for the flow Laplacian on nonhomogeneous trees

2023/10/13 by Alessio Martini, Martini, Alessio, Federico Santagati +5 · 1 citation
Computer Science · Mathematics · #05C05 #05C21 #42B20 #43A99 #Advanced Mathematical Modeling in Engineering #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Spectral Theory in Mathematical Physics #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2310.09113

openalex publication_date 2023/10/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let T be a locally finite tree equipped with a flow measure m. Let \mathcal L be the flow Laplacian on (T,m). We prove that the first order Riesz transform ∇ \mathcal L^-1/2 is bounded on L^p(m) for p∈ (1,∞). Moreover, we prove a sharp L^p spectral multiplier theorem of Mihlin--Hörmander type for \mathcal L. In the case where m is locally doubling, we also prove corresponding weak type and Hardy space endpoint bounds. This generalises results by Hebisch and Steger for the canonical flow Laplacian on homogeneous trees to the setting of nonhomogeneous trees with arbitrary flow measures. The proofs rely on approximation and perturbation arguments, which allow one to transfer to any flow tree a number of L^p bounds that hold on homogeneous trees of arbitrarily large degree and are uniform in the degree.

Cited by

• Harmonic Bergman spaces on locally finite trees

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