On the existence of hyperfunction solutions of linear differential equations of the first order with degenerate real principal symbols
Tetsuji Miwa · Proceedings of the Japan Academy Series A Mathematical Sciences · 1973
0. Introduction.Let P be a first order linear partial differential operator of the form P--,j--1 ax /xi, where (a) e GL(n, R).In this note we study the problem of existence of hyperfunction solutions of the equation Pu--f mainly in the case where n--2, and show that the local solvability is valid in some cases.It is easy to see that P is locally solvable except in the neighborhood of the origin.On the other hand, the principal symbol of P vanishes at the origin.Hence our main interest is in the local solvability of the equation at the origin.By the flabbiness of the sheaf of hyperfunctions we can deduce the local solvability from the global solvability, while we will show later that P_q3(R)--_(Rn) in some cases.