Radial Schur multipliers on trees and quantitative Fourier--Schur separation

We study ordinary and completely bounded norms of radial Schur multipliers on trees. On the homogeneous tree of countably infinite degree, we show that boundedness implies complete boundedness. The argument realizes arbitrary matrix amplifications of a shifted Toeplitz symbol inside rectangular corners of the original multiplier. It also applies to some locally finite trees of unbounded degree. Consequently, the completely bounded radial multiplier theorem of González-Pérez, Parcet, Pérez García and Wang has no general counterpart with ordinary norms. We also determine the optimal order of the comparison constant for positive definite even symbols. For each $1<p<\mathrm{\infty }$ with $p\mathrm{\ne }2$, we construct radial symbols on finitely generated nonabelian free groups, supported in the ball of radius $N$, whose completely bounded Herz–Schur norms remain bounded as $N\to \mathrm{\infty }$, whereas their ordinary Fourier multiplier norms grow at least like $N^{\mathrm{\mid }1-\frac{2}{p}\mathrm{\mid }}$. For each fixed $0<\vartheta <\pi$, we also determine the exact order $t^{-\mathrm{\mid }1-\frac{2}{p}\mathrm{\mid }}$ of both the ordinary and completely bounded Fourier multiplier norms of the damped radial phases $s\mapsto {\mathrm{e}}^{(-t+\mathrm{i}\vartheta )\mathrm{\mid }s\mathrm{\mid }}$ as $t\to 0$. Finally, we compute the local amenability modulus of free balls, we bound the Fourier–Schur ratio of free balls from above and below in terms of this modulus, and we show that its restriction to radial symbols grows polynomially in the radius.