Poisson Transforms for Trees of Bounded Degree

TL;DR

This paper introduces a family of Poisson transforms on bounded degree trees, provides explicit inverses, and links eigenfunction growth to boundary regularity.

Contribution

It presents a new parameterized family of Poisson transforms on trees and characterizes eigenfunction growth via boundary regularity.

Findings

Explicit inverses for the Poisson transforms are constructed.

Eigenfunction growth is characterized by boundary H"older regularity.

The transforms are applicable to trees with bounded degree.

Abstract

We introduce a parameterized family of Poisson transforms on trees of bounded degree, construct explicit inverses for generic parameters, and characterize moderate growth of Laplace eigenfunctions by H\"older regularity of their boundary values.