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

Borderline case of traces and extensions for weighted Sobolev spaces