In the recent years, certain wavelet-type transformations such as the curvelet or shearlet transformation have gained considerable attention, due to their potential for
efficiently handle data with features along edges. In both cases, it was shown that the decay rate of the corresponding transformation coefficients of a tempered distribution identifies the wavefront set of that distribution. Roughly speaking, the wavefront set of a tempered distribution $u$ is the set of points $t\in{\mathbb R}^n$ and directions $\xi\in S^{n-1}$ along which $u$ is not smooth at $t$.

Recently, many efforts have been made aiming to generalize the characterization of the wavefront set of a tempered distribution, in terms of its continuous wavelet transform, for higher dimensional continuous wavelet transforms. In this talk, we consider the problem of characterizing the Sobolev wavefront set of a distribution constructed using square-integrable representations of ${\mathbb R}^n\rtimes H$ where $H$ can be any suitably chosen dilation group. We tackle two important cases: 1) the mother wavelet is compactly supported, and 2) the mother wavelet has compactly supported Fourier transform.

This talk is based on joint work with Hartmut Fuhr.