In 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.

在本文中，我们介绍了新的函数空间，我们称其为各向异性双曲Besov和Triebel-Lizorkin空间。它们的定义基于双曲Littlewood-Paley分析，该分析涉及仅在平滑权重中出现的各向异性矢量。这样的空间提供了一个一般而自然的环境，以便理解可以使用双曲小波（在文献中有时也称为张量积小波）描述什么样的各向异性光滑度，迄今为止，小波类主要用于描述支配空间的特征。混合平滑度。我们当前工作的核心是基于双曲小波变换的这些新空间的表征。在此，我们使用标准的方法来处理这两种方法，即使用具有足够平滑度，衰减度和消失力矩的小波系统，而且还有非常简单和基本的双曲线Haar系统。我们追求的第二个主要问题是新颖的双曲空间与经典各向异性Besov-Lizorkin-Triebel标度之间的关系。如我们的结果所示，通常，两种解决各向异性的方法都不重合。但是，在Sobolev范围内，情况就是如此，它提供了将新获得的双曲小波表征应用到经典设置的链接。特别地，这允许通过通用双曲小波基的系数来检测经典的各向异性，而无需适应各向异性的基或先验知识。如我们的结果所示，通常，两种解决各向异性的方法都不重合。但是，在Sobolev范围内，情况就是如此，它提供了将新获得的双曲小波表征应用到经典设置的链接。特别地，这允许通过通用双曲小波基的系数来检测经典的各向异性，而无需适应各向异性的基或先验知识。如我们的结果所示，通常，两种解决各向异性的方法都不重合。但是，在Sobolev范围内，情况就是如此，它提供了将新获得的双曲小波表征应用到经典设置的链接。特别地，这允许通过通用双曲小波基的系数来检测经典的各向异性，而无需适应各向异性的基或先验知识。