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A Calderón-Zygmund extension theorem for abstract Sobolev spaces. (English) Zbl 0855.46023

Summary: It is shown that if the Boyd indices of the rearrangement invariant Banach function space \(L_\rho (\mathbb{R}^n)\) are strictly between 0 and 1, \(\rho\) is an absolutely continuous function norm, \(\Omega\) is a domain from \(\mathbb{R}^n\) satisfying the restricted cone condition, denoting by \(\omega\) the restriction of \(\rho\) to \(\Omega\), there exists an extension operator for the abstract Sobolev space \(W^m L_\omega (\Omega)\). This is a generalization to abstract Sobolev spaces of a result obtained by de Souza for Orlicz-Sobolev spaces.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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