$L^p$-boundedness of pseudo-differential operators on homogeneous trees

TL;DR

This paper investigates the boundedness of pseudo-differential operators on homogeneous trees, establishing links to integer groups and extending classical theorems to this setting for various p-values.

Contribution

It introduces new results on $L^{p}$-boundedness of pseudo-differential operators on homogeneous trees, including an analogue of the Calderon-Vaillancourt theorem.

Findings

Established $L^{p}$-boundedness for $p eq 2$ on homogeneous trees.

Connected boundedness properties on trees to those on the group of integers.

Proved an analogue of the Calderon-Vaillancourt theorem in this setting.

Abstract

The aim of this article is to study the $L^{p}$-boundedness of pseudo-differential operators on a homogeneous tree $ \mathfrak{X} $. For $p\in (1,2)$, we establish a connection between the $L^{p}$-boundedness of the pseudo-differential operators on $ \mathfrak{X} $ and that on the group of integers $\mathbb{Z}$. We also prove an analogue of the Calderon-Vaillancourt theorem in the setting of homogeneous trees, for $p\in(1,\infty)\setminus\{2\}$.