Traces of weighted function spaces: dyadic norms and Whitney extensions

TL;DR

This paper reviews the characterization of trace spaces of weighted fractional Sobolev spaces using dyadic cube averages, facilitating function extension via Whitney operators.

Contribution

It introduces a new dyadic norm-based characterization of trace spaces for weighted fractional smoothness spaces, aiding in function extension methods.

Findings

Dyadic cube averages effectively characterize trace spaces.

The approach simplifies extending functions with Whitney operators.

Provides a unified framework for weighted fractional Sobolev spaces.

Abstract

The trace spaces of Sobolev spaces and related fractional smoothness spaces have been an active area of research since the work of Nikolskii, Aronszajn, Slobodetskii, Babich and Gagliardo among others in the 1950's. In this paper we review the literature concerning such results for a variety of weighted smoothness spaces. For this purpose, we present a characterization of the trace spaces (of fractional order of smoothness), based on integral averages on dyadic cubes, which is well adapted to extending functions using the Whitney extension operator.