STRUCTURE OF DIFFERENTIAL OPERATORS ON A LOCALLY COMPACT ABELIAN GROUP S. S. Akbarov In [i] Bruhat described a method for endowing any locally compact group G with a smooth structure. A great number of publications have since been devoted to Bruhat's construction and related questions, in view of its many applications in harmonic analysis and probability theory [2-13]. In this paper we present a theorem that gives the general form of a differ- ential operator on an arbitrary locally compact abelian group X. i. Main Theorem and Corollaries. Let X be an arbitrary LCA-group in additive notation with neutral element 0, ~iX} the algebra of smooth functions on X, in Bruhat's sense, with values in R or C. In [14-16] I defined the notion of a homogeneous flow (that i~, a flow along a one-parameter subgroup) and described the properties of the space ~(X) of homogeneous flows in X. I showed, in particular, that any smooth function ~:r~(X) is differentiable along every homogeneous flow u~(X), and moreover the derivative I also pointed out that the topological vector space .~r has a is again a smooth function. basis {e;;ie~il: where the series converges in ~(X) and the coefficients ~R are continuous functions of Following Peetre [17], we define a differential operator on X to be any linear map D: $(X) ~(X) satisfying the locality condition The simplist differential operators on X are the operators of multiplication by functions and the derivations of the algebra ~(X) described in [16]. Define a multi-index of finite order over a set I to be any family k={k~; i~I} of non- negative integers (k~Z+) almost all of which (i.e., all but a finite number) vanish: card {i~l: k~0}<~. The set of all multi-indices of finite order over I will be denoted by M(I). It is a lattice relative to the natural order: k~l-~Vi~I k~l~. The least element of this lattice is the null multi-index o~M(I): V i~I o~=0. We will use the term support of a multi- index k~M(I) for the (finite) set suppk={i~I: k~r Further, we define the order and fac- torial of a multi-index k as k!--~]'Ii~ik~! =:Hi~su,pkki! 9 Fix an arbitrary basis {e~; iE]} in ~(X) ~ [14, 15]. To every nonnull multi-index we can associate a differential operator 0e ~ Ali~supp ~ \ Oe i / (since homogeneous flows commute with one another, the product on the weight is well defined). The null multi-index needs a separate definition: Moscow Institute of Electrical Engineering. Translated from Matematicheskie Zametki, Vol. 52, No. 4, pp. 3-14, October, 1992. Original article submitted May 12, 1991. 0001-4346/92/5234-0995512.50 9 1993 Plenum Publishing Corporation 995