Traces of Sobolev functions with one square integrable directional derivative

Abstract

AbstractWe consider the Sobolev spaces of square integrable functions v, from ℝn or from one of its hyperquadrants Q, into a complex separable Hilbert space, with square integrable sum of derivatives ∑∂𝓁v. In these spaces we define closed trace operators on the boundaries ∂Q and on the hyperplanes {r𝓁 = z}, z ∈ ℝ\{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 ℝn and the regularity properties of its restrictions to the hyperquadrants Q. Copyright © 2005 John Wiley & Sons, Ltd.