Riesz transforms on the Vicsek set
Abstract
We study Riesz and reverse Riesz inequalities for the fractional powers of the Laplacian Δ on the unbounded Vicsek set. The space carries its Hausdorff measure m. The gradient ∂ is a weak gradient defined on the skeleton, and it is measured with respect to a length measure ν that is singular with respect to m. The relevant fractional order for the Riesz inequalities is not 1/2 but γp=(df+p−1)/pdw, where df=log35 and dw=df+1. For p∈[1,2], we prove the weak-type reverse Riesz inequality |(−Δ)γpf|Lp,∞(m)≲|∂ f|Lp(ν). The corresponding strong estimate fails for 1≤ p<2, whereas the weak estimate fails for 2<p<∞, even after heat regularization. For the Riesz transform ℛp=∂(−Δ)−γp, we show that ℛ1 is not of weak type (1,1) but that ℛp is bounded from Lp(m) to Lp,∞(ν) for p∈(1,2), and from Lp,1(m) to Lp(ν) for p∈(2,∞). However, if p ≥ 1, ℛp is bounded from Lp(m) to Lp(ν) only for p=2.
Related
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- REVERSE INEQUALITY FOR THE RIESZ TRANSFORMS ON RIEMANNIAN MANIFOLDS
- Martingale Transforms and Compensated Bellman Estimates for Dunkl Riesz Transforms
- A stochastic Carleson embedding theorem with constant e and the vector of Riesz transforms
- Endpoint Riesz transforms on normalized graphs
BibTeX
@misc{baudoin2026riesz,
title = {{Riesz transforms on the Vicsek set}},
author = {Fabrice Baudoin and Aobo Chen and Li Chen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33858},
eprint = {2609.33858},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33858},
}
RIS
TY - UNPB
TI - Riesz transforms on the Vicsek set
AU - Fabrice Baudoin
AU - Aobo Chen
AU - Li Chen
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33858
UR - https://arxiv.org/abs/2609.33858
AB - We study Riesz and reverse Riesz inequalities for the fractional powers of the Laplacian Δ on the unbounded Vicsek set. The space carries its Hausdorff measure m. The gradient ∂ is a weak gradient defined on the skeleton, and it is measured with respect to a length measure ν that is singular with respect to m. The relevant fractional order for the Riesz inequalities is not 1/2 but γ_p=(d_f+p−1)/pd_w, where d_f=log_35 and d_w=d_f+1. For p∈[1,2], we prove the weak-type reverse Riesz inequality |(−Δ)^γ_pf|_L^p,∞(m)≲|∂ f|_L^p(ν). The corresponding strong estimate fails for 1≤ p<2, whereas the weak estimate fails for 2<p<∞, even after heat regularization. For the Riesz transform ℛ_p=∂(−Δ)^−γ_p, we show that ℛ_1 is not of weak type (1,1) but that ℛ_p is bounded from L^p(m) to L^p,∞(ν) for p∈(1,2), and from L^p,1(m) to L^p(ν) for p∈(2,∞). However, if p ≥ 1, ℛ_p is bounded from L^p(m) to L^p(ν) only for p=2.
ER -