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

By Jon Johnsen, Winfried Sickel. 2008

80 citations recorded by the source corpus.

The subject is traces of Sobolev spaces with mixed Lebesgue norms on Euclidean space. Specifically, restrictions to the hyperplanes given by x1 = 0 and xn = 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 x1 = 0. For x1 the trace spaces are proved to be mixed‐norm Lizorkin–Triebel spaces with a specific sum exponent; for xn 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)

Subfields: Applied Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/666548216d124d9666e2a319f37131e6e7165171

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Citations in the corpus

A structurally damped plate equation with Dirichlet-Neumann boundary conditions (2015, background): “8 in [21] in a more general setting.”
Atomic and Littlewood–Paley Characterizations of Anisotropic Mixed-Norm Hardy Spaces and Their Applications (2018, background): “For more developments of mixed-norm function spaces, we refer the reader to [16, 18, 27, 29, 42, 43].”
General Parabolic Mixed Order Systems in Lp and Applications (2013, methodology): “In the following we want to show that the anisotropic Triebel-Lizorkin spaces with mixed norms used in [JS08] can be represented as an intersection of two spaces in some special cases.”
A direct proof of Sobolev embeddings for quasi-homogeneous Lizorkin-Triebel spaces with mixed norms (2017, unknown): “Hence the fact that θp = p0 gives, by taking Lp -norms in (8), (10) ‖u |F s p,q‖ ≤ c ‖u |F s0 p0,∞‖(c ‖ u |F s0 p0,∞‖) = c′′ ‖u |F s0 p0,∞‖.”
Boundedness of commutators of fractional integral operators on mixed Morrey spaces (2019, background): “Moreover, other spaces with mixed norm, mixed Lorentz spaces [10–12], mixed Besov spaces and Triebel–Lizorkin spaces [13,14], andmixedHardy spaces [15,16] have been studied in recent years.”
Anisotropic Lizorkin–Triebel spaces with mixed norms — traces on smooth boundaries (2015, methodology): “For the reader’s sake, it is recalled that the main purpose of [16] was to extend the classical theory of trace operators to the F a p,q -scales.”
Wavelet transforms for homogeneous mixed-norm Triebel–Lizorkin spaces (2016, methodology): “For contributions on the embedding properties and traces on hyperplanes, the reader is referred to works of Johnsen and Sickel [23, 24], who in collaboration with Munch Hansen also analysed embeddings and equivalent characterisations of such mixed-norm spaces, their invariance under coordinate trans”
Pseudodifferential operators on mixed‐norm Besov and Triebel–Lizorkin spaces (2016, background): “23) We shall need a variation of Fefferman–Stein vector-valued maximal inequality (see [2], [17]): If p = (p1, .”
Traces and embeddings of anisotropic function spaces (2012, unknown): “1) was obtained in [43, 44] for q ≤ p and more general cases were considered in [10, 21].”