WEIERSTRASS DIYISION{ WITH QUASIA|{AIYTIC tsOUNE}ARY VATUES

Let g: R+*R+ (R+ is the set of nonnegative real numbers) be a convex increasing function such that g(0):0 and t-Lg(t)**- as l**-. Define a sequence {M,}n<z * (Z+ is the set of nonnegative integers) by Mn:exp g(n), nQZ+. 'We assume g grows fast enough to ensure that M,>n!, n€Z+. Let Q be a domain in ck with 0 a point on its boundary bQ. we denote by,il1,:s,ilt({M,}, O) the set of germs at 0 of complex-valued Whitney C- functions f on Q (the closure of O) which are analytic in Q and satisfy the following growth conditions on their derivatives: for each r>0 sufficiently small thatf is represented by a function on O nloQ) (lo1r1:{zCCk: lzl-r,1=7=k}) there exist constants A and B, which depend in general on both/and r but not on n€Z+, such that for all nCZ+, (1.1) sup lD.f(t)l = ABnMn.a€(Z+)k,lal=n, z(.Q)Ay(r) (D":D2:6tap*...0fit, where z:(z', z*):(4,.'., zx) are coordinates on Ce, a:(a1,...,d.X(Z+)k, and lol:or+...*ar.) We assume that do is quasianalytic in the sense of Denjoy and Carleman: (1.2) f e Ao and Df (O) : O for all a€(Z+)k imply f:0€dr. Before going on, we remark that by the use of the logarithmic convexity of the sequence {M,), it is not difficult to show that,ilo is a local algebra with maximal ideal mo:{f€.ilo:"f(0):0}. The quasianalyticity assumption is independent of the'dimension k.If the sequence {M"} satisfies certain additional hypotheses, then *y' * is closed under composition whenever the composition makes sense, and./p is also closed under differentiation. For a more complete discussion, see [2]. In this paper we consider a quasianalytic local algebra eo(M"D.we show a weierstrass-Malgrange-Mather type division theorem does not hold in do(M"|) 76 C. L. Cnrrpnnss if k>2,,ilo(nlfi is a proper subset of,il,({M,}), andbeis c2 smooth and strongry Levi pseudoconvex at 0. If, however, bo is Levi pseudoflat at 0, we prove a generic division theorem holds in do({M,l), k>2. wefurther show in this case that division is possible in.ilo(I[,] by every regurar erement of 0o, the local argebra of germs at 0 of analytic functions. (The case in which bo is pseudoconcave at 0 is trivial, since in this case do(M"\) reduces to @0. See L. Hörmander, [4].