PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 129, Number 12, Pages 3531–3537 S 0002-9939(01)06013-0 Article electronically published on June 28, 2001 REGULARITY PROPERTIES OF DISTRIBUTIONS AND ULTRADISTRIBUTIONS S. PILIPOVI ´ C AND D. SCARPALEZOS (Communicated by Jonathan M. Borwein) Abstract. We give necessary and sufficient conditions for a regularized net of a distribution in an open set Ω which imply that it is a smooth function or C k function in Ω. We also give necessary and sufficient conditions for an ultradistribution to be an ultradifferentiable function of corresponding class. Introduction Algebras of generalized functions are usually defined as factor algebras of cer- tain algebras of sequences (nets) of smooth functions (cf. [1], [2], [3], [8], [11]), and the classical spaces of functions, distributions and ultradistributions are em- bedded into the appropriate algebra through their regularizations; they are also equivalence classes of sequences (nets) of smooth functions. Solutions in algebras of generalized functions of PDE are also presented by such sequences (nets) and even differential operators (for example, with singular coefficients) are sometimes replaced by sequences (nets) of differential operators with smooth coefficients. So the natural question arises: Under what conditions is a given generalized function (a generalized solution of PDE) actually a classical function of appropriate class, distribution, or ultradistribution? The partial answers to such questions are given in Propositions 1, 2 and more gen- erally in Theorems 1, 2, respectively, in terms of asymptotic behaviour of sequences of functions provided that these sequences are evaluated on the ultraproduct Ω N . Denote by (θ n ) (respectively, (φ n )) a δ-sequence, also called a sequence of molli- fiers, of smooth functions (respectively, of appropriate ultradifferentiable functions). Precise definitions will be given in sections 2 and 5, respectively. We will prove: Proposition 1. (i) Let (T n ) be a regularized sequence of T ∈E ′ (Ω), i.e. T n = T ∗ θ n ,n ∈ N. If (∃m ∈ R)(∀(x n ) ∈ Ω N )(∀α ∈ N 0 )(T (α) n (x n )= O(n m )), then T ∈ C ∞ 0 (Ω).(O is the Landau symbol.) (ii) Let (T n ) be a regularized sequence of T ∈E ′k (Ω),k ∈ N 0 . If (∀(x n ) ∈ Ω N )(∀α ≤ k)(T (α) n (x n )= O(1)), then T ∈ C k 0 (Ω). Let (M p ) be a sequence of positive numbers satisfying conditions (M.1) ∗ , (M.2) and (M.3)’. These conditions and the definitions of corresponding ultradistribution spaces will be given in section 1. Received by the editors February 21, 2000. 2000 Mathematics Subject Classification. Primary 46F05, 46F30, 03C20. c 2001 American Mathematical Society 3531 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use