Quantitative characterization of traces of Sobolev maps

We give a quantitative characterization of traces on the boundary of Sobolev maps in Ẇ1,(ℳ,), where ℳ and are compact Riemannian manifolds, ℳ≠∅: the Borel-measurable maps :ℳ→ that are the trace of a map ∈Ẇ1,(ℳ,) are characterized as the maps for which there exists an extension energy density :ℳ→[0,∞] that controls the Sobolev energy of extensions from ⌊−1⌋-dimensional subsets of ℳ to ⌊⌋-dimensional subsets of ℳ.