Traces of Sobolev functions with one square integrable directional derivative

We consider the Sobolev spaces of square integrable functions v, from ?ⁿ or from one of its hyperquadrants Q, into a complex separable Hilbert space, with square integrable sum of derivatives ∑?_??v. In these spaces we define closed trace operators on the boundaries ?Q and on the hyperplanes {r_?? = z}, z ∈ ?\{0}, which turn out to be possibly unbounded with respect to the usual L²‐norm for the image. Therefore, we also introduce bigger trace spaces with weaker norms which allow to get bounded trace operators, and, even if these traces are not L², we prove an integration by parts formula on each hyperquadrant Q. Then we discuss surjectivity of our trace operators and we establish the relation between the regularity properties of a function on ?ⁿ and the regularity properties of its restrictions to the hyperquadrants Q. Copyright © 2005 John Wiley & Sons, Ltd.