TY - JOUR

T1 - Linear extension operators for Sobolev spaces on radially symmetric binary trees

AU - Fefferman, Charles

AU - Klartag, Bo'Az

N1 - Funding Information:
Funding information : Charles Fefferman was supported by the Air Force Office of Scientific Research (Grant number FA9950-18-1-0069) and the National Science Foundation (NSF) (Grant number DMS-1700180). Bo’az Klartag was supported by a grant from the Israel Science Foundation (ISF).
Publisher Copyright:
© 2023 the author(s), published by De Gruyter.

PY - 2023/1/1

Y1 - 2023/1/1

N2 - 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.

AB - 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.

KW - binary tree

KW - linear extension operator

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U2 - 10.1515/ans-2022-0075

DO - 10.1515/ans-2022-0075

M3 - Article

AN - SCOPUS:85164184172

SN - 1536-1365

VL - 23

JO - Advanced Nonlinear Studies

JF - Advanced Nonlinear Studies

IS - 1

M1 - 20220075

ER -