Singular differential operators and distributions

Differential operatorsp(t, ∂)=a _m (t)∂ ^m +···+a ₀(t), wherea _m has a zero of finite order att=0, are studied as operators on the distribution spacesD'(R) andE'(R). In particular the kernel ofp, operating onD'(R), is studied in detail by use of asymptotic analysis and a simple formula for its dimension is given. A continuous right inverse forp onD'(R) is constructed. Necessary and sufficient conditions for this inverse to be two-sided are given. Extensions are made to the spacesE (R) andE'(R). Finally some features for operators with more than one singular point are briefly discussed and there is noted a phenomenon — forced propagation of supports — which has important consequences in higher dimensions as a forced propagation of singularities.