ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 16, Number 3, Summer 1986 HYPERBOLIC OPERATORS IN SPACES OF GENERALIZED DISTRIBUTIONS SALEH ABDULLAH Hyperbolic operators were investigated by L. Ehrenpreis [4] in the space of Schwartz distributions and by C. C. Chou [3] in the spaces of Roumieu ultradistributions. In this paper we study hyperbolic operators in spaces of Beurling generalized distributions. (See [1] and [2]). Let D', E\ D' w , Ea be the spaces of distribution, distributions with compact support, generalized distributions and generalized distributions with compact support in R M , respectively. DEFINITION. The convolution operator S, S e E' oe , is said to be co- hyperbolic with respect to / > 0 (resp. / < 0) if there exists a fundamental solution ir + (resp. E~), E + ; E~ e D'^ so that supp E + a {(*, t) e R n x R: t ^ - b 0 + bi\x\} for some b 0 ,bi > 0(resp. supp E~ a {(x, t) eR n x R: t <* b 0 — bi \x\} for some £ 0 , b\ > 0). An operator is said to be co-hyperbolic if it is co-hyperbolic with res- pect to / > 0 and t < 0. This definition coincides with the definition of hyperbolicity introduced by Ehrenpreis [4, Theorem 2] for Schwartz dis- tributions. For the notation and the properties of generalized distributions we refer to [2]. Let co e Jt c (see [2, Definition 1.3.23]). Using Proposition 1.2.1 of [2] we could extend oe to C n without losing any of its original properties; we will assume that o is the extended function. We use the estimate (0 o>(0 = o(|?/log If I), as |f | -> oo, from which it follows that (2) o)(£) è M(ì + |£|), for some constant M. Following Ehrenpreis we prove the following theorem which char- acterizes co-hyperbolic operators. The theorem and its proof will be given in the case of co-hyperbolicity with respect to t > 0, the other case could be proved similarly. Received by the editors on February 23, 1984. Copyright © 1986 Rocky Mountain Mathematics Consortium 535