The problem of the uniqueness in the Cauchy problem for linear dierential operators has been widely investigated during the last years (see [Z] for references). It is now well understood in the analytic framework, with Holmgren's theorem, where uniqueness always holds (at least for non characteristic surfaces) and in the C1 case, with HoE rmander's theorem ([H1], IV, chap. 28) where the uniqueness is governed by principal normality and pseudo-convexity. The purpose of this work is to ®ll the gap between these two theorems by considering operators with C1 and partly analytic coecients. In particular one of our results will contain both the theorems mentioned above. Let us be more precise. Let na, nb be two non negative integers with n na nb 1. We shall set Rn Rna Rnb and, for x or n in Rn, x xa; xb, n na; nb. Let P P x;D P xa; xb;Dxa ;Dxb be a linear dierential operator of arbitrary order m, with principal symbol pm. We shall assume that