Riesz transform on eventually Gaussian local trees
Abstract
We study the Riesz transform ℛ=∂(−Δ)−1/2 on uniform local trees, metric measure spaces that are locally real trees and whose canonical Dirichlet form is built from weak derivatives along the skeleton. The reference measure m may be singular with respect to the length measure ν, so boundedness of ℛ is understood from Lp(m) to Lp(ν). In contrast with fractal-like manifolds and cable systems, the diffusion is sub-Gaussian at small scales and Gaussian at large scales. Under uniform volume growth, two-sided heat kernel estimates and a pointwise gradient estimate for the heat kernel, we prove that a local Dini condition on the scale function implies boundedness of ℛ on Lp for every p∈[2,∞), and hence the reverse Riesz inequality for every p∈(1,2]. Conversely, boundedness of ℛ for some p<2, or a reverse Riesz inequality for some p>2, forces the space to be one-dimensional at small scales. We show that a reverse Hölder inequality for harmonic functions yields the gradient estimate, and verify all hypotheses for spaces carrying a geometric group action whose generators have bounded displacement. As an application, for the alternating Vicsek fractafold in ℤd we determine the exact ranges of p for which the Riesz and reverse Riesz inequalities hold.
Related
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- Riesz transform characterization of Hardy spaces associated with Dunkl--Schrödinger operators
- Riesz transform on manifolds with mixed ends for 1<q<2
BibTeX
@misc{baudoin2026riesz,
title = {{Riesz transform on eventually Gaussian local trees}},
author = {Fabrice Baudoin and Aobo Chen and Li Chen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.34331},
eprint = {2609.34331},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34331},
}
RIS
TY - UNPB
TI - Riesz transform on eventually Gaussian local trees
AU - Fabrice Baudoin
AU - Aobo Chen
AU - Li Chen
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.34331
UR - https://arxiv.org/abs/2609.34331
AB - We study the Riesz transform ℛ=∂(−Δ)^−1/2 on uniform local trees, metric measure spaces that are locally real trees and whose canonical Dirichlet form is built from weak derivatives along the skeleton. The reference measure m may be singular with respect to the length measure ν, so boundedness of ℛ is understood from L^p(m) to L^p(ν). In contrast with fractal-like manifolds and cable systems, the diffusion is sub-Gaussian at small scales and Gaussian at large scales. Under uniform volume growth, two-sided heat kernel estimates and a pointwise gradient estimate for the heat kernel, we prove that a local Dini condition on the scale function implies boundedness of ℛ on L^p for every p∈[2,∞), and hence the reverse Riesz inequality for every p∈(1,2]. Conversely, boundedness of ℛ for some p<2, or a reverse Riesz inequality for some p>2, forces the space to be one-dimensional at small scales. We show that a reverse Hölder inequality for harmonic functions yields the gradient estimate, and verify all hypotheses for spaces carrying a geometric group action whose generators have bounded displacement. As an application, for the alternating Vicsek fractafold in ℤ^d we determine the exact ranges of p for which the Riesz and reverse Riesz inequalities hold.
ER -