Triebel–Lizorkin spaces and shearlets on the cone in R2

The shearlets are a special case of the wavelets with composite dilation that, among other things, have a basis-like structure and multi-resolution analysis properties. These relatively new representation systems have encountered wide range of applications, generally surpassing the performance of their ancestors due to their directional sensitivity. Both theories of coorbit spaces and decomposition spaces provide a way of associating some kind of smoothness spaces to shearlets. However, these smoothness spaces are closer to classical Besov type spaces. Here, instead, we define a kind of highly anisotropic inhomogeneous Triebel–Lizorkin spaces and prove that it can be characterized with the so-called “shearlets on the cone” coefficients. We first prove the boundedness of the analysis and synthesis operators with the “traditional” shearlets coefficients. Then, with the development of the smooth Parseval frames of shearlets of Guo and Labate we are able to prove a reproducing identity, which was previously possible only for the L2 case. We also find some embeddings of the (classical) dyadic spaces into these highly anisotropic spaces, and vice versa, for certain ranges of parameters. In order to keep a concise document we develop our results in the “weightless” case (w=1) and give hints on how to develop the weighted case.