Borderline case of traces and extensions for weighted Sobolev spaces

TL;DR

This paper characterizes the existence of traces and extensions for weighted Sobolev spaces at borderline weights, establishing connections with new Besov-type spaces on hyperplanes.

Contribution

It provides a complete characterization of trace spaces and introduces new Besov-type spaces related to weighted Sobolev spaces at critical weights.

Findings

Full characterization of trace space existence

Development of Besov-type spaces on hyperplanes

Analysis of trace and extension relations at borderline weights

Abstract

In this paper, we study the traces and the extensions for weighted Sobolev spaces on upper half spaces when the weights reach to the borderline cases. We first give a full characterization of the existence of trace spaces for these weighted Sobolev spaces, and then study the trace parts and the extension parts between the weighted Sobolev spaces and a new kind of Besov-type spaces (on hyperplanes) which are defined by using integral averages over selected layers of dyadic cubes.