Shearlets and pseudo-differential operators

Abstract

Shearlets on the cone are a multi-scale and multi-directional discrete system that have near-optimal representation of the so-called cartoon-like functions. They form Parseval frames, have better geometrical sensitivity than traditional wavelets and an implementable framework. Recently, it has been proved that some smoothness spaces can be associated to discrete systems of shearlets. Moreover, there exist embeddings between the classical isotropic dyadic spaces and the shearlet generated spaces. We prove boundedness of pseudo-differential operators (PDO’s) with non regular symbols on the shear anisotropic inhomogeneous Besov spaces and on the shear anisotropic inhomogeneous Triebel–Lizorkin spaces (which are up to now the only Triebel–Lizorkin-type spaces generated by either shearlets or curvelets and more generally by any parabolic molecule, as far as we know). The type of PDO’s that we study includes the classical Hormander definition with x-dependent parameter $$\delta $$ for a range limited by the anisotropy associated to the class. One of the advantages is that the anisotropy of the shearlet spaces is not adapted to that of the PDO.