Linear extension operators for Sobolev spaces on radially-symmetric binary trees

TL;DR

This paper constructs a linear extension operator for Sobolev spaces on radially-symmetric binary trees, ensuring minimal Sobolev norm extension of functions from leaves to the entire tree, with bounds depending only on p.

Contribution

It introduces a new harmonic extension operator tailored for weighted binary trees, extending Sobolev functions with a norm bound depending solely on p, unlike standard operators.

Findings

Existence of a bounded linear extension operator for Sobolev spaces on binary trees.

The operator is a variant of harmonic extension, adapted to tree weights and structure.

Boundedness depends only on the Sobolev exponent p.

Abstract

Let $1 < p < \infty$ and suppose that we are given a function $f$ defined on the leaves of a weighted tree. We would like to extend $f$ to a function $F$ defined on the entire tree, so as to minimize the weighted $W^{1,p}$-Sobolev norm of the extension. An easy situation is when $p = 2$, where the harmonic extension operator provides such a function $F$. In this note we record our analysis of the particular case of a radially-symmetric binary tree, which is a complete, finite, binary tree with weights that depend only on the distance from the root. Neither the averaging operator nor the harmonic extension operator work here in general. Nevertheless, we prove the existence of a linear extension operator whose norm is bounded by a constant depending solely on $p$. This operator is a variant of the standard harmonic extension operator, and in fact it is harmonic extension with respect to a certain Markov kernel determined by $p$ and by the weights.