Linear extension operators for Sobolev spaces on radially symmetric binary trees

Charles Fefferman, Bo'Az Klartag

Research output: Contribution to journalArticlepeer-review

Abstract

Let 1 < p < ∞ 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 F defined on the entire tree, so as to minimize the weighted W1, p-Sobolev norm of the extension. An easy situation is when p = 2, where the harmonic extension operator provides such a function F 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 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.

Original languageEnglish (US)
Article number20220075
JournalAdvanced Nonlinear Studies
Volume23
Issue number1
DOIs
StatePublished - Jan 1 2023

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • General Mathematics

Keywords

  • binary tree
  • linear extension operator

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