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

Abstract

The subject is traces of Sobolev spaces with mixed Lebesgue norms on Euclidean space. Specifically, restrictions to the hyperplanes given by the first and last coordinates 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 except for the borderline cases; these are also covered in case of the first variable. With respect to the first variable the trace spaces are proved to be mixed-norm Lizorkin--Triebel spaces with a specific sum exponent. For the last variable they are similarly defined Besov spaces. The treatment includes continuous right-inverses and higher order traces. The results rely on a sequence version of Nikolskij's inequality, Marschall's inequality for pseudo-differential operators (and Fourier multiplier assertions), as well as dyadic ball criteria.

32 pages. This is the accepted version, with a few unimportant misprints corrected. Appeared in Mathmatische Nachrichten in 2008

References in corpus (2)

• A direct proof of Sobolev embeddings for quasi-homogeneous Lizorkin--Triebel spaces with mixed norms
• Traces of Besov spaces revisited

Cited by in corpus (7)

• Wavelet transforms for homogeneous mixed-norm Triebel--Lizorkin spaces
• Anisotropic Lizorkin--Triebel Spaces with Mixed Norms --- Traces on Smooth Boundaries
• Characterisation by Local Means of Anisotropic Lizorkin--Triebel Spaces with Mixed Norms
• Anisotropic, Mixed-Norm Lizorkin--Triebel Spaces and Diffeomorphic Maps
• $L_p$-Theory of Type $1,1$-Operators
• Fundamental Results for Pseudo-Differential Operators of Type $1,1$
• Atomic and Littlewood-Paley Characterizations of Anisotropic Mixed-Norm Hardy Spaces and Their Applications