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Including the previously untreated borderline cases, the trace spaces in thedistributional sense of the Besov--Lizorkin--Triebel spaces are determined forthe anisotropic (or quasi-homogeneous) version of these classes. The ranges ofthe trace are in all cases shown to be approximation spaces, and these areshown to be different from the usual spaces precisely in the previouslyuntreated cases. To analyse the new spaces, we carry over some realinterpolation results as well as the refined Sobolev embeddings of J.~Frankeand B.~Jawerth to the anisotropic scales.

Traces of anisotropic Besov--Lizorkin--Triebel spaces---a complete treatment of the borderline cases