Lifting of Quaternionic Frames to Higher Dimensions with Partial Ridges

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.