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Endpoint Riesz transforms on normalized graphs
Abstract
For an infinite connected locally finite undirected graph with positive symmetric edge weights and normalized Laplacian Δ, we prove weak type (1,1) estimates for two Riesz transforms. The edge transform has universal constant $2$, with no volume assumption. The vertex carré-du-champ transform has constant H1+1 whenever H1=supx m(B(x,1))/m(x)<∞. These endpoint constants apply to real-valued inputs and yield strong ℓp bounds for 1<p≤2. The proof uses a discrete obstacle decomposition obtained by a contraction mapping, followed by elementary support estimates for the two gradients.
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BibTeX
@misc{wang2026endpoint,
title = {{Endpoint Riesz transforms on normalized graphs}},
author = {Zihao Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28531},
eprint = {2609.28531},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28531},
}
RIS
TY - UNPB
TI - Endpoint Riesz transforms on normalized graphs
AU - Zihao Wang
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.28531
UR - https://arxiv.org/abs/2609.28531
AB - For an infinite connected locally finite undirected graph with positive symmetric edge weights and normalized Laplacian Δ, we prove weak type (1,1) estimates for two Riesz transforms. The edge transform has universal constant $2$, with no volume assumption. The vertex carré-du-champ transform has constant H_1+1 whenever H_1=sup_x m(B(x,1))/m(x)<∞. These endpoint constants apply to real-valued inputs and yield strong ℓ^p bounds for 1<p≤2. The proof uses a discrete obstacle decomposition obtained by a contraction mapping, followed by elementary support estimates for the two gradients.
ER -