In spaces with a slow diffusion, the Riesz transform is unbounded on L^p, p∈ (2,∞)

2025/02/15 by Joseph Feneuil, Feneuil, Joseph
Computer Science · Mathematics · #42B20 #43A85 #60J10 #60J60 #Digital Filter Design and Implementation #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Mathematical and Theoretical Analysis #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2502.10837

openalex publication_date 2025/02/15 · openalex created_date 2025/02/19 · openalex updated_date 2026/07/28

Abstract

In graphs and Riemannian manifolds where the kernel of the diffusion semigroup satisfies pointwise sub-Gaussian estimates, we study the range of parameters \( p ∈ (1, ∞) \) and \( γ∈ [0, 1] \) for which the quantities \( ‖Δ^γf‖_p \) and \( ‖∇ f‖_p \) can be compared. In particular, we prove that in such metric spaces, the Riesz transform \( ∇ Δ^-1/2 \) is unbounded on \( L^p \) for all \( p ∈ (2, ∞) \), thereby demonstrating a clear departure from the behavior observed in the Euclidean setting.

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