Resumen de Extension criteria for homogeneous Sobolev spaces of functions of one variable

Pavel Shvartsman

• For each p>1 and each positive integer m, we give intrinsic characterizations of the restriction of the homogeneous Sobolev space Lmp(R) to an arbitrary closed subset E of the real line. We show that the classical one-dimensional Whitney extension operator is "universal" for the scale of Lmp(R) spaces in the following sense: For every p∈(1,∞], it provides almost optimal Lmp-extensions of functions defined on E. The operator norm of this extension operator is bounded by a constant depending only on m. This enables us to prove several constructive Lmp-extension criteria expressed in terms of m-th order divided differences of functions.