Linear extension operators for Sobolev spaces on radially symmetric binary trees

Abstract

Let 1<p<∞1\lt p\lt \infty and suppose that we are given a function ff defined on the leaves of a weighted tree. We would like to extend ff to a function FF defined on the entire tree, so as to minimize the weighted W1,p{W}^{1,p}-Sobolev norm of the extension. An easy situation is when p=2p=2, where the harmonic extension operator provides such a function FF. 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 pp. 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 pp and by the weights.

Keywords

• linear extension operator
• binary tree
• 46e36