A note on properties of the restriction operator on Sobolev spaces

In our companion paper [3] we studied a number of different Sobolev spaces on a general (non-Lipschitz) open subset Ω of Rn, defined as closed subspaces of the classical Bessel potential spaces Hs(Rn) for s ϵ R. These spaces are mapped by the restriction operator to certain spaces of distributions on Ω. In this note we make some observations about the relation between these spaces of global and local distributions. In particular, we study conditions under which the restriction operator is or is not injective, surjective and isometric between given pairs of spaces.We also provide an explicit formula for minimal norm extension (an inverse of the restriction operator in appropriate spaces) in a special case.

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Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).