The Conditions for Some Linear Partial Differential Equations to Be Solvable in G

Abstract

Using Bargmann's transformation and some basic results of theory of analytic functions with several complex variables, we have disscussed two classes of LPDOs in this paper. We prove that each operator of one class of them is surjective both from G to G and from L² to L², but not injective, and each operator of another class is injective from G to G but not surjective. And in the letter case, the necessary and suffcient conditions for the corresponding equations to be solvable in G are given.