The Diophantine nature for the convergence of formal solutions
Masafumi Yoshino · Tohoku Mathematical Journal · 1986
In this paper we shall study the diophantine nature of the problem of the convergence of all formal solutions.Concerning the convergence of all formal solutions, Kashiwara-Kawai-Sjδstrand [1] studied the equation Pu = Σuri=ι*ιs» a a β(x)x a (d/dx) β u = / and gave a sufficient condition for the convergence of all formal solutions.Unfortunately this condition is merely sufficient and not necessary.As for the necessity few results are known.This is mainly because we must treat rather delicate problems of diophantine nature.Concerning this, the first work which clearly showed the diophantine nature of the problem of the convergence of formal solutions was perhaps that of SiegeΓs in [4].On the other hand in 1974, Leray [2] studied the diophantine nature of the Goursat problem by using a new diophantine function p.Though the problems they studied seem to be quite different, their basic ideas are closely connected.More precisely, their methods to treat the diophantinetype difficulty are the same.In this paper we shall introduce two diophantine functions σ ξ and p which are generalizations of SiegeΓs condition in [4] and Leray's auxiliary function in [2], respectively.By using these functions we shall give necessary and sufficient conditions for the convergence of formal solutions.We remark that this yields the solvability of the same equation by the usual method.We also give examples showing that we cannot drop any of the assumptions of the main theorem in general.Finally, we point out that the method here is also applicable to the study of C°° (or C ω )hypoellipticity of operators on the torus by slight modification.The author would like to give sincere thanks to the referee and the editor who kindly gave the author many usefull suggestions in preparing this paper.2. Notation and results.Let x = (x lf x 2 ) be the variable in C 2 .For η 6 R 2 and a multi-index a = (a lf a 2 ) eN 2 , N = {0,1, 2, •}, we set η a = ηpηp and (x-d)° = (xAY ι (xAY 2 , where 3 = (3 lf 3 2 ) and dj = d/toj (j = 1, 2).Let m ^ 1 be an integer and let co eCV Then we are concerned with the