Besides assertions on sharp embeddings of B^s_{pq} and F^s_{pq} we give necessary sufficient conditions the parameters s,p,q,p_1,q_1,p_2,q_2 for Hölder type inequalities \| f \cdot g | ≤ c B^s_{p_1q_1} B^s_{p_2q_2} F^s_{p_1q_1} F^s_{p_2q_2} to hold.

The subject is traces of Sobolev spaces with mixed Lebesgue norms on Euclidean space. Specifically, restrictions to the hyperplanes given by first and last coordinates are applied functions belonging quasi-homogeneous, mixed-norm Lizorkin--Triebel spaces; obtained from these as special cases. Spaces admitting in distribution sense characterised except for borderline cases; also covered case variable. With respect variable trace proved be a specific sum exponent. For they similarly defined...

This article deals with trace operators on anisotropic Lizorkin–Triebel spaces mixed norms over cylindrical domains smooth boundary. As a preparation we include rather self-contained exposition of manifolds and extend these results to mixed-norm cylinders in Euclidean space. In addition Rychkov's universal extension operator for half space is shown be bounded respect the norms, support preserving right-inverse given explicitly proved continuous scale spaces. an application, heat equation...

Abstract We investigate the asymptotic behaviour of entropy numbers compact embedding $B^{s_1}_{p_1,q_1}(\mathbb{R}^d,w_1)\hookrightarrow B^{s_2}_{p_2,q_2}(\mathbb{R}^d,w_2)$. Here $B^s_{p,q}(\mathbb{R}^d,w)$ denotes a weighted Besov space. present general approach which allows us to work with large class weights.

We characterize the set of all functions f \mathbb R to itself such that associated superposition operator T_f: g \to \circ maps class BV^1_p (\mathbb R) into itself. Here , 1 \le p < \infty denotes primitives bounded -variation, endowed with a suitable norm. It turns out an is always and sublinear. Also, consequences for boundedness operators defined on Besov spaces B^s_{p,q}({\mathbb R}^n) are discussed.

Abstract We investigate the rate of convergence interpolating splines with respect to sparse grids for Besov spaces dominating mixed smoothness (tensor product spaces). Main emphasis is given approximation by piecewise linear functions. Keywords: gridsrate convergenceSobolev–Besov smoothnesstensor spaceswavelet decompositionsapproximation from hyperbolic crossestensor splinesWhittaker's cardinal seriesAMS Subject Classifications:: 41A2541A6342B9946E3565D0565D07

The article deals with a simplified proof of the Sobolev embedding theorem for Lizorkin–Triebel spaces (that contain L p ‐Sobolev as special cases). method extends to corresponding fact general based on mixed ‐norms. In this context Nikol′ skij–Plancherel–Polya inequality sequences functions satisfying geometric rectangle condition is proved. results extend also anisotropic quasi‐homogeneous type.

In this paper, we compare the recent approach of Hans Triebel to introduce smoothness spaces related Morrey‐Campanato with Besov type and Triebel‐Lizorkin spaces. These two scales have been introduced some years ago represent a further variant measure by using Morrey

If \Omega is a bounded domain in \mathbb R^n , 1 ≤ q < p \infty and s=0, 1, 2,\ldots then we clearly have W^{s,p}(\Omega)\subset W^{s,q}(\Omega) . We prove that this property does not hold when s an integer.

Abstract In this article, the authors determine optimal regularity of characteristic functions in Besov-type and Triebel–Lizorkin-type spaces under restrictions on measure $$\delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>δ</mml:mi> </mml:math> -neighborhoods boundary. particular, necessary sufficient conditions for membership these snowflake domain also some spiral type domains are obtained.