We consider the technique of lifting frames to higher dimensions with the ridge idea that originally was introduced by Grafakos and Sansing. We pursue a novel approach with regard to a non-commutative setting, concretely the skew-field of quaternions. Moreover, we allow for splitting dimensions and for lifting with regard to multi-ridges. To this end, we introduce quaternionic Sobolev spaces and prove the corresponding embedding theorems. We mention as concrete examples quaternionic wavelet frames and quaternionic shearlet frames, and give the respective lifted families.

中文翻译：

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将四元离子框架提升到带有部分凸脊的更高尺寸

我们考虑采用最初由Grafakos和Sansing引入的“山脊”概念将框架提升到更高尺寸的技术。我们针对非交换环境，特别是四元数的偏场，寻求一种新颖的方法。此外，我们允许拆分尺寸和提升多脊。为此，我们介绍了四元数的Sobolev空间，并证明了相应的嵌入定理。作为具体示例，我们提到四元小波框架和四元小波框架，并给出各自的提升族。