Characteristic Cauchy problems in the complex domain

Abstract

Gårding-Kotake-Leray showed that in a certain characteristic Cauchy problem

$$Pu = \upsilon \in o\;(the\;sheaf\;of\;holomorphic\;functions)$$

with zero Cauchy data on a hypersurface S, u can be ramified. Moreover, u is of the form

$$(*)\;\upsilon (x) = \sum\limits_{i = 0}^{q - 1} {\upsilon i(x){{[k(x)]}^{1/q}}} $$

where q is a positive integer ≥ 2 and u is ramified around K : k(x) = 0. Here K is tangent to S at characteristic points of S. Let us denote by $N_{q,K}^m$ the class of functions which have the form (*) and whose first m traces on S vanish.