  We define distribution spaces of a sequence of convolutions of a set of
distributions with smooth functions, the shearlet system. Then, we define
associated sequence spaces and prove characterizations. We also show a
reproducing identity in the class of distributions. Finally, we prove
Sobolev-type embeddings within the shear anisotropic inhomogeneous spaces and
embeddings between (classical dyadic) isotropic inhomogeneous spaces and shear
anisotropic inhomogeneous spaces.
