$L^p$ spherical multipliers on homogeneous trees

TL;DR

This paper characterizes $L^p$ spherical multipliers on homogeneous trees for all $p$ except 2, linking them to $L^p$ Fourier multipliers on the torus, thus extending the understanding of harmonic analysis on trees.

Contribution

It provides a complete characterization of $L^p$ spherical multipliers on homogeneous trees in terms of Fourier multipliers on the torus, for all relevant $p$ values.

Findings

Characterization of $L^p$ spherical multipliers on homogeneous trees.

Connection established between tree multipliers and torus Fourier multipliers.

Extension of harmonic analysis results to a broader class of $p$ values.

Abstract

We characterise, for each $p$ in $[1,\infty) \setminus \{2\}$, the class of $L^p$ spherical multipliers on homogeneous trees in terms of $L^p$ Fourier multipliers on the torus.