Propagation of analytic and Gevrey singularities for operators with non- involutive characteristics
Massimo Cicognani, Luisa Zanghirati · Kyoto journal of mathematics · 1993
In this paper we cosider a class of analytic operators with multiple non-involutive characteristics and study the propagation of analytic and Gevrey singularities.To state precisely our result, we begin by recalling that ffor a constant C independent on a E Z + n .We denote by G d (X ) the space of all f E C -(X ) which are of class G d at every xoE X , and write Go d (X ) forFor d >1, the spaces of d-ultradistributions G ( d ) '( X ), Go ( d ) "(X ) are the dual spaces of G d (X ) and Go d (X ) respectively and G ( d ) "(X ) can be identified with the space of all elements of Go ( d ) "(X ) with compact support.We recall also that the space D '(X ) of all distributions in X can be identified with a subspace of Go ( d Y (X ) for a ll d > 1 .If l1, is defined as follows: for a fixed (xo, eo)E T*(X )\0, we say that (xo, eo)qWFd(f) if there exist Go d (X) with 0(x)=1 in a neighborhood of xo, and positive constants C, E such that the Fourier transform (çbf)^ of Of satisfies:(1.2) 1 (sbf)"( )1 1: let us consider a sequence {q5,1c Go d (X ) , q5,(x)=1 in a neighborhood of xo, such that there exists a constant C and for every 6 >0 a consotant Cc satisfying