On the Ranges of Realizations in Distribution Spaces

The paper deals with the closedness of ranges and the surjectivity of realizations related to linear partial differential operators L(x,D). A characterization (with the help of a certain coercivity condition) of the surjectivity of the maximal realization L’’ k, G in BP, k( G ) c D'(Ci )I ti for some 1' E B ,, , k)is established. Here 2pk (p £ ( 0, 1), k € K )is the Hdrmander space. Furthermore, the closedness of the range R(A,k(G)) corresponding to the minimal realization A k( 0) in local i-Iörmander spaces G) = U € D'(G )I 41 si £ Bp , k for any 'I' £ C°(G )} is considered.