Conormal Singularities

Abstract

The simplest functions having singularities are not the ones with prescribed wave front set, constructed out of the example given in (1.3). The natural building blocks to take are the classical distributions with nontrivial singular support; for example, the Dirac distribution ∂₀(x) the Heaviside function H(x _i ), or their smoothed versions |x|^s, (x _i ),^r,|x _i |^r, and so on. An appropriate class of functions singular across the hypersurface x _i =0, for example, would be one containing H(x _i ) and allowing for multiplication by smooth functions. We are naturally led to the notion of conormal distribution, considered in great generality in Hörmander [37].