A direct obstacle proof of a dimension-free weak-type (1,1) bound for the Gaussian Riesz transforms
Abstract
We give a detailed, direct proof of a dimension-free weak-type (1,1) estimate for the first-order Gaussian vector Riesz transform, using the fractional obstacle method of Ouyang, Spector, and Stockdale. For dγn=π−n/2e−|x|2 dx, let L=−1/2Δ+x·∇, D=2−1/2∇, and Rγ=DL−1/2Π0, where Π0 removes the mean. We prove that Rγ has weak-type constant at most $2$ on real-valued inputs, independently of n. The main construction is the unique unshifted decomposition F=g+L1/2u, for nonnegative F∈ L2(γn) and a>∫ F dγn, with u≥0, min{F,a}≤ g≤ a, preservation of mass, and g=a on {u>0}. We retain the details of coercivity, the Lewy--Stampacchia estimate, the upgrade from the form domain to the operator domain, and locality. We also identify the L1 extension with the principal-value transform and obtain the constant 8/π for complex-valued inputs.
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BibTeX
@misc{gutierrez2026direct,
title = {{A direct obstacle proof of a dimension-free weak-type $(1,1)$ bound for the Gaussian Riesz transforms}},
author = {Cristian E. Gutiérrez and Juan Manfredi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26478},
eprint = {2610.26478},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26478},
}
RIS
TY - UNPB
TI - A direct obstacle proof of a dimension-free weak-type (1,1) bound for the Gaussian Riesz transforms
AU - Cristian E. Gutiérrez
AU - Juan Manfredi
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26478
UR - https://arxiv.org/abs/2610.26478
AB - We give a detailed, direct proof of a dimension-free weak-type (1,1) estimate for the first-order Gaussian vector Riesz transform, using the fractional obstacle method of Ouyang, Spector, and Stockdale. For dγ_n=π^−n/2e^−|x|^2 dx, let L=−1/2Δ+x·∇, D=2^−1/2∇, and R_γ=DL^−1/2Π_0, where Π_0 removes the mean. We prove that R_γ has weak-type constant at most $2$ on real-valued inputs, independently of n. The main construction is the unique unshifted decomposition F=g+L^1/2u, for nonnegative F∈ L^2(γ_n) and a>∫ F dγ_n, with u≥0, min{F,a}≤ g≤ a, preservation of mass, and g=a on {u>0}. We retain the details of coercivity, the Lewy--Stampacchia estimate, the upgrade from the form domain to the operator domain, and locality. We also identify the L^1 extension with the principal-value transform and obtain the constant 8/π for complex-valued inputs.
ER -