Schur Multipliers and Spherical Functions on Homogeneous Trees

TL;DR

This paper characterizes Schur multipliers on homogeneous trees and derives explicit formulas for their norms, with applications to Fourier multipliers on free groups and spherical functions on p-adic groups.

Contribution

It provides a necessary and sufficient condition for radial functions to be Schur multipliers on homogeneous trees and computes their norms explicitly.

Findings

Necessary and sufficient condition for Schur multipliers on homogeneous trees.

Explicit formula for the Schur norm of radial functions.

Closed expression for Fourier multiplier norms on free groups and spherical functions on p-adic groups.

Abstract

Let X be a homogeneous tree of degree q+1 (for q between 2 and infinity) and let f be a complex function on X times X for which f(x,y) only depend on the distance between x and y in X. Our main result gives a necessary and sufficient condition for such a function to be a Schur multiplier on X times X. Moreover, we find a closed expression for the Schur norm of f. As applications, we obtain a closed expression for the completely bounded Fourier multiplier norm of the radial functions on the free (non-abelian) group on N generators (for N between 2 and infinity) and of the spherical functions on the p-adic group PGL_2(Q_q) for every prime number q.