Some directional microlocal classes defined using wavelet transforms

In this short paper we discuss how the position - scale half-space of wavelet analysis may be cut into different regions. We discuss conditions under which they are independent in the sense that the T\"oplitz operators associated with their characteristic functions commute modulo smoothing operators. This shall be used to define microlocal classes of distributions having a well defined behavior along lines in wavelet space. This allows us the description of singular and regular directions in distributions. As an application we discuss elliptic regularity for these microlocal classes for domains with cusp-like singularities.