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

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 $p\in (1,\infty )\setminus \{2\}$.