PreprintCC BY 4.0arXiv
The Sharp Scale Inequality for the Schatten Trace Class
Abstract
In this paper, we establish the metric cotype $2$ inequality for the Schatten trace class S1 at the optimal order of the scaling parameter. More precisely, for every integer n≥ 1, every even integer m≥ 4, and every mapping f:ℤmn→ S1, the following holds &(∑j=1n𝔼x|f(x+m/2ej)−f(x)|12)1/2 ≤&(3√(70) m+6√n)(𝔼x,ε|f(x+ε)−f(x)|12)1/2, where x and ε are independently uniform on ℤmn and {−1,0,1}n, respectively. The order √n is necessary for any fixed metric cotype constant. This settles the optimal scaling problem for the Schatten trace class posed by Mendel and Naor \cite[Section 8]{MN} (see also \cite[Question 3]{GMN}).
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BibTeX
@misc{jiao2026sharp,
title = {{The Sharp Scale Inequality for the Schatten Trace Class}},
author = {Yong Jiao and Sijie Luo and Lian Wu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21445},
eprint = {2610.21445},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21445},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Sharp Scale Inequality for the Schatten Trace Class
AU - Yong Jiao
AU - Sijie Luo
AU - Lian Wu
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21445
UR - https://arxiv.org/abs/2610.21445
AB - In this paper, we establish the metric cotype $2$ inequality for the Schatten trace class S_1 at the optimal order of the scaling parameter. More precisely, for every integer n≥ 1, every even integer m≥ 4, and every mapping f:ℤ_m^n→ S_1, the following holds &(∑_j=1^n𝔼_x|f(x+m/2e_j)−f(x)|_1^2)^1/2 ≤&(3√(70) m+6√n)(𝔼_x,ε|f(x+ε)−f(x)|_1^2)^1/2, where x and ε are independently uniform on ℤ_m^n and {−1,0,1}^n, respectively. The order √n is necessary for any fixed metric cotype constant. This settles the optimal scaling problem for the Schatten trace class posed by Mendel and Naor \cite[Section 8]{MN} (see also \cite[Question 3]{GMN}).
ER -