# The function classes $\gamma_H(\beta, \delta, d)$ and global linear Goursat problems

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## Abstract

The function classes H(, , d) and global linear Goursat problems ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA Classe di Scienze JAN PERSSON The function classes γH(β,δ,d) and global linearGoursat problems Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 3e série, tome 24, no 4 (1970), p. 633-639. <http://www.numdam.org/item?id=ASNSP_1970_3_24_4_633_0> © Scuola Normale Superiore, Pisa, 1970, tous droits réservés. L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ THE FUNCTION CLASSES 03B3H(03B2, 03B4, d) AND GLOBAL LINEAR GOURSAT PROBLEMS (*) JAN PFRSSON Introduction. The non-characteristic linear Cauchy problem when the coefficients are entire functions was treated in [4] and [5] as special cases of more general problems. It was proved that with entire data the Cauchy problem has an entire solution if the coefficients in the principal part of the operator are constants. In [4] it was pointed out that the solution of is Since u is not an entire function we must put some restriction on the co- efficients in the principal part. Recent studies on the characteristic Cauchy problem for the equation I q ~ 0, 1 q, m, n integers, 1 see [6], show that the dependence on the space variable has a remarkable impact on the solutions. See also Asadullin [1], and A. Friedman [2]. It was shown in [6] that the only analytic solutions around the origin of are the trivial ones Ax + B, A and B being arbitrary con- stant. It was further shown in [6] that Dt u + tx3

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