Carleson measures on locally finite trees

TL;DR

This paper characterizes Carleson measures on locally finite trees, linking them to Poisson integral boundedness and exploring their relation to BMO functions on the boundary for trees with bounded degree.

Contribution

It provides a new characterization of Carleson measures on trees and connects these measures to Poisson integrals and boundary BMO functions.

Findings

Carleson measures characterized via Poisson integral boundedness

Established connection between Carleson measures and BMO functions on boundary

Results apply specifically to trees with bounded degree

Abstract

We provide a characterization of Carleson measures on locally finite trees. This characterization establishes the connection between Carleson measures and the boundedness of a suitable Poisson integral between $L^p$-spaces. Additionally, when the tree has bounded degree, we investigate the relationship between Carleson measures and BMO functions defined on the boundary of the tree.