Pseudo-differential calculus on homogeneous trees

TL;DR

This paper develops a pseudo-differential calculus on homogeneous trees to analyze eigenfunctions of the discrete Laplacian, providing tools for studying their concentration and oscillation properties.

Contribution

It introduces a new pseudo-differential calculus framework on homogeneous trees, including symbol classes, operator bounds, and formulas for adjoints, products, and commutators.

Findings

Operators are bounded on L^2

Derived formulas for adjoint and product of operators

Computed the symbol of the commutator with the Laplacian

Abstract

In the objective of studying concentration and oscillation properties of eigenfunctions of the discrete Laplacian on regular graphs, we construct a pseudo-differential calculus on homogeneous trees, their universal covers. We define symbol classes and associated operators. We prove that these operators are bounded on L^2 and give adjoint and product formulas. Finally we compute the symbol of the commutator of a pseudo-differential operator with the Laplacian.