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
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))
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