On the trace problem for Lizorkin–Triebel spaces with mixed norms

Abstract

Abstract The subject is traces of Sobolev spaces with mixed Lebesgue norms on Euclidean space. Specifically, restrictions to the hyperplanes given by x 1 = 0 and x n = 0 are applied to functions belonging to quasi‐homogeneous, mixed norm Lizorkin–Triebel spaces ; Sobolev spaces are obtained from these as special cases. Spaces admitting traces in the distribution sense are characterised up to the borderline cases; these are also covered in case x 1 = 0. For x 1 the trace spaces are proved to be mixed‐norm Lizorkin–Triebel spaces with a specific sum exponent; for x n they are similarly defined Besov spaces. The treatment includes continuous right‐inverses and higher order traces. The results rely on a sequence version of Nikol'skij's inequality, Marschall's inequality for pseudodifferential operators (and Fourier multiplier assertions), as well as dyadic ball criteria. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)