Trace Operators on Regular Trees

Pekka Koskela,Khanh Ngoc Nguyen,Zhuang Wang

doi:10.1515/agms-2020-0117

Pekka Koskela, Khanh Ngoc Nguyen + Show 1 more

Open Access

https://doi.org/10.1515/agms-2020-0117

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Abstract

AbstractWe consider different notions of boundary traces for functions in Sobolev spaces defined on regular trees and show that the almost everywhere existence of these traces is independent of the chosen definition of a trace.

Highlights

Let us begin with the classical setting
We consider di erent notions of boundary traces for functions in Sobolev spaces de ned on regular trees and show that the almost everywhere existence of these traces is independent of the chosen de nition of a trace
Towards a more constructive de nition of a trace, let us extend u to a function Eu ∈ W, (Rn). This is possible by classical extension theorems in [5, 24]

Summary

Introduction

Let us begin with the classical setting. Exists for almost every ξ ∈ ∂Bn( , ). Almost everywhere refers to the surface measure on ∂Bn( , ). In this sense, u has a well de ned trace almost everywhere on ∂Bn( , ). Towards a more constructive de nition of a trace, let us extend u to a function Eu ∈ W , (Rn). This is possible by classical extension theorems in [5, 24]. By the version of Lebesgue di erentiation theorem for Sobolev functions [26, Section 5.14], the limitlim r→ mn(B(x, r))

Eu dmn

Note that

For any