Inhomogeneous shearlet coorbit spaces

In this paper we establish inhomogeneous coorbit spaces related to the continuous shearlet transform and the weighted Lebesgue spaces $L_{p,v}, p\geq 1,$ for certain weights $v$. We present an inhomogeneous shearlet frame for $L_2(\mathbb{R}^d)$ which gives rise to a reproducing kernel $R_\mathfrak{F}$ that is not contained in the space $\mathcal{A}_{1,m_v}$. To show that the inhomogeneous shearlet coorbit spaces are Banach spaces we introduce a generalization of the approach of Fornasier, Rauhut and Ullrich.