Traces of Sobolev functions with one square integrable directional derivative

We consider the Sobolev spaces of square integrable functions, from R^n or from one of its hyperquadrants Q, into a complex separable Hilbert space, with square integrable sum of derivatives . In these spaces we define closed trace operators on the boundaries of Q and on the hyperplanes with one fixed coordinate. These trace operators turn out to be possibly unbounded with respect to the usual L^2-norm for the image. Therefore we introduce also bigger trace spaces with weaker norms which allow to get bounded trace operators, and, even if these traces are not L^2, 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 R^n and the regularity properties of its restrictions to the hyperquadrants Q.