Wave factorization

Abstract

It was clearly shown in the previous section that, in the theory of boundary value problems for elliptic pseudodifferential equations which was constructed by M.I.Vishik and G.I.Eskin, the key role was played the symbol factorization of elliptic pseudodifferential operators on one of variables with the other variables fixed. Depending on factorization index one needs to consider the different boundary value problems reflecting the singularities of appeared situation. When studying the pseudodifferential equation in cone (angle) it proves that analogous role will play the wave factorization related to exit into compex domain with respect to all variables at once. Such factorization permits one to use a multivariable variant of the Wiener-Hopf method and to obtain for a pseudodifferential operator in angle the same solvability picture which we had in the half-space case. This approach has been applied to some classes of differential operators too, but as of now it has not been useful. In this section we will introduce the concept of wave factorization for symbols of pseudodifferential operators and we will verify that the set of symbols which admit the wave factorization is large enough.