Riesz transforms for bounded Laplacians on graphs

2017/08/18 by Li Chen, Chen, Li, Thierry Coulhon +3 · 1 citation
Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG) #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1708.05476

openalex publication_date 2017/08/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study several problems related to the ℓ^p boundedness of Riesz transforms for graphs endowed with so-called bounded Laplacians. Introducing a proper notion of gradient of functions on edges, we prove for p∈(1,2] an ℓ^p estimate for the gradient of the continuous time heat semigroup, an ℓ^p interpolation inequality as well as the ℓ^p boundedness of the modified Littlewood-Paley-Stein functions for all graphs with bounded Laplacians. This yields an analogue to Dungey's results in [Dungey08] while removing some additional assumptions. Coming back to the classical notion of gradient, we give a counterexample to the interpolation inequality hence to the boundedness of Riesz transforms for bounded Laplacians for 1

Cited by

• L^p-Boltzmann-Gibbs principle via Littlewood-Paley-Stein inequality

Related

• Riesz transform for 1 ≤ p ≤ 2 without Gaussian heat kernel bound
• Riesz transform on graphs under subgaussian estimates
• A gradient estimate for positive functions on graphs
• Laplacians on infinite graphs: Dirichlet and Neumann boundary conditions
• Unbounded Laplacians on Graphs: Basic Spectral Properties and the Heat\n Equation