  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.
