Dyadic Norm Besov-Type Spaces as Trace Spaces on Regular Trees

Pekka Koskela,Zhuang Wang

doi:10.1007/s11118-019-09808-5

Abstract

In this paper, we study function spaces defined via dyadic energies on the boundaries of regular trees. We show that correct choices of dyadic energies result in Besov-type spaces that are trace spaces of (weighted) first order Sobolev spaces.

Highlights

Over the past two decades, analysis on general metric measure spaces has attracted a lot of attention, e.g., [2, 4, 12, 13, 15,16,17]
Sobolev spaces, Besov spaces and TriebelLizorkin spaces on metric measure spaces have been studied in [5, 25, 26] via hyperbolic fillings
A related approach was used in [23], where the trace results of Sobolev spaces and of related fractional smoothness function spaces were recovered by using a dyadic norm and the Whitney extension operator

Summary

Introduction

Over the past two decades, analysis on general metric measure spaces has attracted a lot of attention, e.g., [2, 4, 12, 13, 15,16,17]. It is natural to ask whether the Besov-type space Bpθ,λ(∂X) in Definition 2.12 defined via the Bpθ,λ-energy is a trace space of a suitable Sobolev space defined on the regular tree. For any p = (β − log K)/ ≥ 1, there exists a bounded nonlinear extension operator E : Lp(∂X) → N 1,p(X) so that the trace operator T defined via limits of E(f ) along geodesic rays for f ∈ Lp(∂X) satisfies T ◦ E = Id on Lp(∂X). For p = 1 and λ > 0, the optimal space is B1α(∂X) rather than B10,λ by Theorem 1.3 This splitting happens since the two extension operators from Theorems 1.3 and 1.4 are very different: the latter one is of Whitney type in the sense that the extension to an edge is based on the average of the boundary function over the dyadic “shadow” of size comparable to that of the edge, while the former one uses the average over a dyadic boundary element for the definition of the extension to several edges of different sizes. The notation A B (A B) means that there is a constant C such that A ≤ C · B (A ≥ C · B)

Regular Trees and Their Boundaries

Newtonian Spaces on X

Proofs