Forum Math. 18 (2006), 789–801 DOI 10.1515/FORUM.2006.039 Forum Mathematicum ( de Gruyter 2006 Equalities in algebras of generalized functions Stevan Pilipovic ´, Dimitris Scarpalezos, Vincent Valmorin (Communicated by Joram Lindenstrauss) Abstract. Distributional and strong equalities in Colombeau algebra G of generalized functions are compared. Also it is done for G y . Moreover a positive answer to a question of M. Ober- guggenberger is given: A generalized function, invariant under all translations, is a generalized constant. 2000 Mathematics Subject Classification: 46F30, 46F05, 46S10. 1 Introduction The algebra of Colombeau generalized functions GðWÞ is adequate for the analysis of various classes of non linear partial di¤erential equations (cf. [1], [2], [3], [7]) or even for linear partial di¤erential equations with singular coe‰cients which do not have solutions in the setting of the distribution theory (cf. [5] for the linear theory of Colombeau generalized functions). This algebra has many surprising properties, for example, an element can have generalized point values equal zero at any point of W but not being equal zero ([4]). Thus an analogy with the classical function theory is always far from trivial. In this paper we prove some new structural properties of Colombeau generalized functions. Our main tool is the use of Baire theorem as well as the use of parametrix. Our main results are the following. a) A generalized function invariant under all translations is a generalized constant (Theorem 6). It was a conjecture of M. Oberguggenberger. b) If f A G y ðWÞ and Ð W f r dt ¼ 0, in C, for every r A C y 0 ðWÞ, then f ¼ 0 (Theo- rem 5). Thus, the distribution equality of elements in G y ðWÞ implies their strong equality. c) Let f ; g A GðWÞ. Though ð W f r dt ¼ ð W gr dt; in C; for every r A C y 0 ðWÞ; ð * Þ does not imply f ¼ g in GðWÞ, it is true if ( * ) holds for every r A C k 0 ðWÞ, for some k (Theorem 4). Brought to you by | California Institute of Technology Authenticated Download Date | 5/27/15 2:15 AM