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Register a new userOn the trace problem for Triebel--Lizorkin spaces with mixed norms
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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 Nikolskijs inequality, Marschalls inequality for pseudo-differential operators (and Fourier multiplier assertions), as well as dyadic ball criteria.
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The article deals with a simplified proof of the Sobolev embedding theorem for Lizorkin--Triebel spaces (that contain the $L_p$-Sobolev spaces $H^s_p$ as special cases). The method extends to a proof of the corresponding fact for general Lizorkin--Triebel spaces based on mixed $L_p$-norms. In this context a Nikolskij--Plancherel--Polya inequality for sequences of functions satisfying a geometric rectangle condition is proved. The results extend also to spaces of the quasi-homogeneous type.
We prove the continuous dependence of the solution maps for the Euler equations in the (critical) Triebel-Lizorkin spaces, which was not shown in the previous works(cite{Ch02, Ch03, ChMiZh10}). The proof relies on the classical Bona-Smith method as cite{GuLiYi18}, where similar result was obtained in critical Besov spaces $B^1_{infty,1}$.
In this article the authors study complex interpolation of Sobolev-Morrey spaces and their generalizations, Lizorkin-Triebel-Morrey spaces. Both scales are considered on bounded domains. Under certain conditions on the parameters the outcome belongs to the scale of the so-called diamond spaces.
Including the previously untreated borderline cases, the trace spaces in the distributional sense of the Besov--Lizorkin--Triebel spaces are determined for the anisotropic (or quasi-homogeneous) version of these classes. The ranges of the trace are in all cases shown to be approximation spaces, and these are shown to be different from the usual spaces precisely in the previously untreated cases. To analyse the new spaces, we carry over some real interpolation results as well as the refined Sobolev embeddings of J.~Franke and B.~Jawerth to the anisotropic scales.
We characterize the Schauder and unconditional basis properties for the Haar system in the Triebel-Lizorkin spaces $F^s_{p,q}(Bbb R^d)$, at the endpoint cases $s=1$, $s=d/p-d$ and $p=infty$. Together with the earlier results in [10], [4], this completes the picture for such properties in the Triebel-Lizorkin scale, and complements a similar study for the Besov spaces given in [5].
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