Refinement of the Relations between Mixed Smoothness Moduli in $ L_{1} $ and $ L_{q} $ Metrics

2021 ◽  
Vol 62 (4) ◽  
pp. 661-677
Author(s):  
M. K. Potapov ◽  
B. V. Simonov
Keyword(s):  
Mixed Smoothness

scholarly journals Hyperbolic Wavelet Analysis of Classical Isotropic and Anisotropic Besov–Sobolev Spaces

2021 ◽  
Vol 27 (3) ◽  
Author(s):  
Martin Schäfer ◽  
Tino Ullrich ◽  
Béatrice Vedel
Keyword(s):  
Haar System ◽  
A Priori ◽  
Natural Setting ◽  
The Novel ◽  
Major Question ◽  
Vanishing Moments ◽  
Classical Setting ◽  
Hyperbolic Wavelets ◽  
The Relationship ◽  
Mixed Smoothness

AbstractIn this paper we introduce new function spaces which we call anisotropic hyperbolic Besov and Triebel-Lizorkin spaces. Their definition is based on a hyperbolic Littlewood-Paley analysis involving an anisotropy vector only occurring in the smoothness weights. Such spaces provide a general and natural setting in order to understand what kind of anisotropic smoothness can be described using hyperbolic wavelets (in the literature also sometimes called tensor-product wavelets), a wavelet class which hitherto has been mainly used to characterize spaces of dominating mixed smoothness. A centerpiece of our present work are characterizations of these new spaces based on the hyperbolic wavelet transform. Hereby we treat both, the standard approach using wavelet systems equipped with sufficient smoothness, decay, and vanishing moments, but also the very simple and basic hyperbolic Haar system. The second major question we pursue is the relationship between the novel hyperbolic spaces and the classical anisotropic Besov–Lizorkin-Triebel scales. As our results show, in general, both approaches to resolve an anisotropy do not coincide. However, in the Sobolev range this is the case, providing a link to apply the newly obtained hyperbolic wavelet characterizations to the classical setting. In particular, this allows for detecting classical anisotropies via the coefficients of a universal hyperbolic wavelet basis, without the need of adaption of the basis or a-priori knowledge on the anisotropy.

The Jawerth–Franke Embedding of Spaces with Dominating Mixed Smoothness

2009 ◽  
Vol 16 (4) ◽  
pp. 667-682
Author(s):  
Markus Hansen ◽  
Jan Vybíral
Keyword(s):  
Function Spaces ◽  
Dominating Mixed Smoothness ◽  
Mixed Smoothness

Abstract We give a proof of the Jawerth embedding for function spaces with dominating mixed smoothness of Besov and Triebel–Lizorkin type where 0 < 𝑝0 < 𝑝1 ≤ ∞ and 0 < 𝑞0,𝑞1 ≤ ∞ and with If 𝑝1 < ∞, we prove also the Franke embedding Our main tools are discretization by a wavelet isomorphism and multivariate rearrangements.

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