Radial Schur multipliers on trees and the failure of Fourier-Schur transference
Abstract
Given 1 < p < ∞ and a function m : N0 → C, we provide a characterization of radial Schur multipliers M(x,y) = m(d(x,y)) on the Schatten p-class of a tree (X,d) equipped with its graph metric. That is, the operator norm of the radial Schur multiplier SM is controlled by the norm of the Fourier multiplier Tm on the torus: | SM : Sp(X) → Sp(X) |cb ≲ p4/((p−1)2) | Tm : Lp(T) → Lp(T) |cb, with both quantities being comparable when the tree X has infinite diameter. This is a rather unexpected inequality, which necessarily fails at the endpoints p=1,∞ due to the work of Haagerup, Steenstrup and Szwarc. As an application, we construct the first examples of completely Sp-bounded Herz-Schur multipliers in the free group that are unbounded as Fourier multipliers. This result disproves Fourier-Schur transference---a longstanding open problem since the work of Neuwirth and Ricard in 2011. In line with this, our main result also implies the surprising fact that free balls Schur multipliers are uniformly Sp-bounded, in sharp contrast to their Fourier behavior. In addition, we disprove Fourier-Schur transference over the algebraic groups PGL2(Qq) and PSL2(R) with K-biinvariant counterexamples, the latter involving additional geometric and probabilistic techniques.
Related
- Radial Schur multipliers on trees and quantitative Fourier--Schur separation
- Finite abelian groups are Schur multipliers of finite solvable groups
- Weak (1,1) Shur multipliers and the Hörmander--Mikhlin--Schur theorem
- Characterisations for coarse embeddability via quasi-locality and Schur multipliers
- Schur Multipliers with Unequal Operator and Completely Bounded Norms on Sp, $1<p\ne 2<\infty $
BibTeX
@misc{gonzalezperez2026radial,
title = {{Radial Schur multipliers on trees and the failure of Fourier-Schur transference}},
author = {Adrian González-Pérez and Javier Parcet and Jorge Pérez García and Simeng Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15881},
eprint = {2610.15881},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15881},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Radial Schur multipliers on trees and the failure of Fourier-Schur transference
AU - Adrian González-Pérez
AU - Javier Parcet
AU - Jorge Pérez García
AU - Simeng Wang
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15881
UR - https://arxiv.org/abs/2610.15881
AB - Given 1 < p < ∞ and a function m : N_0 → C, we provide a characterization of radial Schur multipliers M(x,y) = m(d(x,y)) on the Schatten p-class of a tree (X,d) equipped with its graph metric. That is, the operator norm of the radial Schur multiplier S_M is controlled by the norm of the Fourier multiplier T_m on the torus: | S_M : S_p(X) → S_p(X) |_cb ≲ p^4/((p−1)^2) | T_m : L_p(T) → L_p(T) |_cb, with both quantities being comparable when the tree X has infinite diameter. This is a rather unexpected inequality, which necessarily fails at the endpoints p=1,∞ due to the work of Haagerup, Steenstrup and Szwarc. As an application, we construct the first examples of completely S_p-bounded Herz-Schur multipliers in the free group that are unbounded as Fourier multipliers. This result disproves Fourier-Schur transference---a longstanding open problem since the work of Neuwirth and Ricard in 2011. In line with this, our main result also implies the surprising fact that free balls Schur multipliers are uniformly S_p-bounded, in sharp contrast to their Fourier behavior. In addition, we disprove Fourier-Schur transference over the algebraic groups PGL_2(Q_q) and PSL_2(R) with K-biinvariant counterexamples, the latter involving additional geometric and probabilistic techniques.
ER -