ALMOST AUTOMORPHIC GENERALIZED FUNCTIONS

Novi Sad J. Math. Vol. 45 No. 1 2015 207-214 ALMOST AUTOMOPHIC GENEALIZED FUNCTIONS Chikh Bouzar 1 Mohammed Taha Khalladi 2 and Fatima Zohra Tchouar 3 Abstract. The paper deals with a new algebra of generalized functions. This algebra contains Bochner almost automorphic functions and almost automorphic distributions. Properties of this algebra are studied. AMS Mathematics Subject Classification (2010): 43A60 46F30 42A75 Key words and phrases: Almost automorphic functions; Almost automorphic distributions; Almost automorphic generalized functions 1. Introduction The concept of almost automorphy is a generalization of Bohr almost periodicity it has been introduced by S. Bochner see [1] and [2]. For a general study of almost automorphic functions see Veech s paper [7]. There is a considerable amount of papers and books on almost periodic functions and also almost automrphic functions. L. Schwartz introduced and studied in [6] almost periodic distributions. The study of almost automorphic Schwartz distributions is done in the work [4]. An algebra of generalized functions containing Bohr almost periodic functions as well as Schwartz almost periodic distributions has been introduced and studied in [3]. In this work we introduce and study a new algebra of generalized functions containing not only Bochner almost automorphic functions and almost automorphic distributions but also the algebra of almost periodic generalized functions of [3]. So naturally this paper can be seen as a continuation of our works on almost periodic generalized functions and almost automorphic distributions. 2. egular almost automorphic functions We consider functions and distributions defined on the whole space. Denote by C b the space of bounded and continuous complex valued functions on endowed with the norm. L of uniform convergence on the space (C b. L ) is a Banach algebra. For the definition and properties of almost automorphic functions see [1] [2] and [7]. 1 University of Oran 1 Department of Mathematics Laboratoire d Analyse Mathématique et Applications. Oran Algeria e-mail: ch.bouzar@gmail.com; bouzar@univ-oran.dz 2 University of Adrar. Adrar Algeria e-mail: ktaha2007@yahoo.fr 3 University of Oran 1 Department of Mathematics Laboratoire d Analyse Mathématique et Applications. Oran Algeria e-mail: fatima.tchouar@gmail.com

208 Chikh Bouzar Mohammed Taha Khalladi Fatima Zohra Tchouar Definition 1. A complex-valued function f defined and continuous on is called almost automorphic if for any sequence of real numbers (s n ) n N one can extract a subsequence (s nk ) k such that and g (x) := lim f (x + s n k ) exists for every x k + lim g (x s n k ) = f (x) for every x. k + Denote by C aa the space of almost automorphic functions on. emark 1. The space C aa is a Banach subalgebra of C b. emark 2. The space of Bohr almost periodic functions is denoted by C ap. Every Bohr almost periodic function is an almost automorphic function and we have C ap C aa C b. Let p [1 + ] and recall the Fréchet space D L p = {φ C : j Z + φ (j) L p} endowed with the countable family of norms φ kp = φ (j) L k Z +. p j k Definition 2. The space of almost automorphic infinitely differentiable functions on denoted by B aa is } B aa := {φ C : j Z + φ (j) C aa. Example 1. The space B ap of regular almost periodic functions see [3] is a Frechet subalgebra of B aa. Some properties of B aa are summarized in the following proposition. Proposition 1. 1. B aa is a subalgebra of C aa. 2. B aa is a Frechet subalgebra of D L. 3. B aa = C aa D L. 4. B aa L 1 B aa. Proof. See [4]. A consequence of Proposition 1 is the following result. Corollary 1. Let u D L then the following statements are equivalent : (i) u B aa. (ii) u φ C aa φ D.

Almost automorphic generalized functions 209 3. Almost automorphic distributions The spaces of L p -distributions introduced in [6] and denoted by D L p are the topological dual spaces of D L q with 1 p + 1 q = 1 and 1 q < +. In particular D L is the topological dual of the space B defined as the closure in 1 D L of the space of smooth functions with compact support. A distribution in D L is called an integrable distribution and a distribution in D 1 L is called a bounded distribution. L. Schwartz provided the following characterization of L p -distributions. Proposition 2. Let p [1 + ]. A tempered distribution T if and only if there exists (f j ) j k L p such that belongs to D L p (3.1) T = k j=0 f (j) j. A study of almost automorphic Schwartz distributions is done in the work [4]. The following result gives characterizations of almost automorphic distributions. Theorem 1. Let T D L T is said to be an almost automorphic distribution if it satisfies one of the following equivalent statements : 1. T φ C aa φ D. 2. (f j ) j k C aa T = j k f (j) j. Definition 3. Denote by B aa the space of almost automorphic distributions. Example 2. The space B ap of almost periodic distributions of Schwartz is a proper subspace of B aa. Some properties of B aa are summarized in the following proposition. Proposition 3. 1. If T B aa then i Z + T (i) B aa. 2. B aa B aa B aa. 3. B aa D L 1 B aa. Proof. See [4]. 4. Almost automorphic generalized functions Let I = ]0 1] and recall the algebra of bounded generalized functions denoted by G L where G L := M L N L

210 Chikh Bouzar Mohammed Taha Khalladi Fatima Zohra Tchouar and M L := N L := { (u ϵ ) ϵ (D L ) I k Z + m Z + u ϵ k = O ( ϵ m) } ϵ 0 { } (u ϵ ) ϵ (D L ) I k Z + m Z + u ϵ k = O (ϵ m ) ϵ 0 emark 3. See [5] for the references on the introduction and the study of the algebras G L p constructed on the Banach spaces L p. Definition 4. The space of almost automorphic moderate elements is defined as { M aa := (u ϵ ) ϵ (B aa ) I k Z + m Z + u ϵ k = O ( ϵ m) } ϵ 0 and the space of almost automorphic negligible elements by { } N aa := (u ϵ ) ϵ (B aa ) I k Z + m Z + u ϵ k = O (ϵ m ) ϵ 0. The main properties of M aa and N aa are given in the following proposition. Proposition 4. 1. The space M aa is a subalgebra of (B aa ) I. 2. The space N aa is an ideal in M aa. Proof. 1. Easy by the results on the algebra B aa see Propostion 1. 2. Let (w ϵ ) ϵ M aa i.e. k Z + m 0 Z + c 0 > 0 ϵ 0 I ϵ < ϵ 0 w ϵ k < c 0 ϵ m 0 and (v ϵ ) ϵ N aa i.e. k Z + m Z + c 1 > 0 ϵ 1 I ϵ < ϵ 1 v ϵ k < c 1 ϵ m. By using the Leibniz formula we find c k > 0 such that w ϵ v ϵ k c k w ϵ k v ϵ k c k c 0 c 1 ϵ m 0+m. Take m Z + such that m 0 + m = m 1 Z + so we obtain k Z + m 1 Z + C = c 0 c 1 c k > 0 ϵ 2 = inf (ϵ 0 ϵ 1 ) I ϵ < ϵ 2 which gives (w ϵ v ϵ ) ϵ N aa. w ϵ v ϵ k < Cϵ m 1 Following the well-known classical construction of algebras of generalized functions of Colombeau type see [5] we introduce the algebra of almost automorphic generalized functions.

Almost automorphic generalized functions 211 Definition 5. The algebra of almost automorphic generalized functions is defined as the quotient G aa := M aa N aa. Notation 1. If u G aa then u = [(u ϵ ) ϵ ] = (u ϵ ) ϵ + N aa where (u ϵ ) ϵ representative of u. emark 4. The algebra of almost automorphic generalized functions G aa is embedded into G L canonically. The following characterization of elements of G aa is similar to the result of Theorem 1-(1). Proposition 5. Let u = [(u ϵ ) ϵ ] G L. Then the following statements are equivalent 1. u G aa. 2. u ε φ B aa ε I φ D. Proof. If u = [(u ϵ ) ϵ ] G aa then u ϵ B aa ϵ I and due to (4) of Proposition 1 u ϵ φ B aa ϵ I φ D. Conversely let u = [(u ϵ ) ϵ ] G L and u ϵ φ B aa ϵ I φ D so u ϵ D L ϵ I and u ϵ φ B aa ϵ I φ D Corollary 1 gives that u ϵ B aa ϵ I. Since u G L we have k Z + m Z + u ϵ k = O ( ϵ m) ϵ 0 consequentely (u ϵ ) ϵ M aa and thus u G aa. emark 5. The characterization 2. from the previous Proposition does not depend on representatives. The following result is easy to prove. Proposition 6. The algebra of almost periodic generalized functions G ap of [3] is embedded canonically into G aa. The following result is well-known. Lemma 1. There exists ρ S satisfying (4.1) ρ (x) dx = 1 and x k ρ (x) dx = 0 k 1. Denote by Σ the set of of functions ρ S satisfying (4.1) and define ρ ϵ (.) := 1 ϵ ρ (. ϵ) ϵ > 0. By means of convolution with a mollifier from Σ we embed the space of almost automorphic distributions B aa into the algebra G aa. is a

212 Chikh Bouzar Mohammed Taha Khalladi Fatima Zohra Tchouar Proposition 7. Let ρ Σ the map i aa : B aa G aa u (u ρ ϵ ) ϵ + N aa is a linear embedding which commutes with derivatives. Proof. Let u B aa then (f j ) j p C aa and u = j p f (j) j. Let us show that (u ρ ϵ ) ϵ M aa. By (4) of Proposition 1 u ρ ϵ B aa ϵ I. Moreover we have ) (u (i) ρ ϵ (x) 1 ϵ i+j f j (x ϵy) ρ (i+j) (y) dy j p then ) sup (u (i) ρ ϵ (x) 1 x ϵ i+j f j L j p consequently C > 0 such that ρ (i+j) (y) dy u ρ ϵ k C ϵ k+p. So (u ρ ϵ ) ϵ M aa. The linearity of i aa follows from the linearity of convolution. If (u ρ ϵ ) ϵ N aa then lim u ρ ϵ = 0 in D ϵ 0 L but it is easy to see that lim u ρ ϵ = u in D ϵ 0 L so u = 0 which means that i aa is injective. Finally ( i ) aa u (j) = [( ) ] u (j) ρ ϵ = [(u ρ ϵ ϵ ϵ) ϵ ] (j) = (i aa (u)) (j). Defining the canonical embedding σ aa : B aa G aa f (f) ϵ + N aa we have two ways to embed the space B aa into G aa by i aa and also by σ aa. The following result gives that we have the same result. Proposition 8. The following diagram commutes. B aa B aa σ aa i aa G aa Proof. It suffices to show that for f B aa we have (f ρ ϵ f) ϵ N aa. Applying Taylor s formula we obtain m Z + ) m (f (j) ρ ϵ (x) f (j) ( ϵy) k (x) = f (k+j) (x) ρ (y) dy + k! k=1 ( ϵy) m+1 (m + 1)! f (m+j+1) (x θ (x) ϵy) ρ (y) dy.

Almost automorphic generalized functions 213 Since f B aa and ρ Σ then f (j) ρ ϵ f (j) L f (m+j+1) consequentely k Z + m = m + 1 i.e. (f ρ ϵ f) ϵ N aa. L f ρ ϵ f k = O(ϵ m ) ε 0 y m+1 ρ ϵ m+1 L 1 (m + 1)! The algebra of tempered generalized functions on C is denoted by G T see [5] for the definition and properties of G T. Proposition 9. Let u = [(u ϵ ) ϵ ] G aa is a well-defined element of G aa. F u := [(f ϵ u ϵ ) ϵ ] + N aa and F = [(f ϵ ) ϵ ] G T then Proof. Since (f ϵ ) ϵ M T and (u ϵ ) ϵ M aa by the classical result of composition of almost automorphic function with continuous function we have f ϵ u ϵ B aa ϵ I. The estimates k Z + m Z + f ϵ u ϵ k = O(ϵ m ) ε 0 are obtained from the fact that (u ϵ ) ϵ M aa and (f ϵ ) ϵ is polynomially bounded. It is easy to prove that the composition is independent on representatives. The convolution of an almost automorphic distribution with an integrable distribution is an almost automorphic distribution. We extend this result to the case of almost automorphic generalized functions. Proposition 10. Let u = [(u ϵ ) ϵ ] G aa and v D L 1 then the convolution u v defined by u v := [(u ϵ v) ϵ ] is a well defined element of G aa. Proof. The characterization (3.1) of elements of D L gives that there exists 1 (f j ) j p L 1 such that v = f (i) i. Let (u ϵ ) ϵ M aa be a representative of i p u. Then u ϵ B aa ϵ I by Proposition 1u ϵ v = u (i) ϵ f i B aa ϵ I. i p Moreover by Young inequality we have (u ϵ v) (j) L f i L 1 u (i+j) L i p ϵ so the fact that (u ϵ ) ϵ M aa gives that k Z + m Z + u ϵ v k = O(ϵ m ) ε 0 consequently (u ϵ v) ϵ M aa. Finally one shows that the result is independent on representatives by obtaining the same estimates.

214 Chikh Bouzar Mohammed Taha Khalladi Fatima Zohra Tchouar We give an extension of the classical Bohl-Bohr theorem. First we recall the definition of a primitive of a generalized function. Definition 6. Let u = [(u ϵ ) ϵ ] G aa and x 0 a primitive of u is a generalized function U defined by U (x) = x x 0 u ϵ (t) dt + N [C] Proposition 11. A primitive of an almost automorphic generalized function is almost automorphic if and only if it is a bounded generalized function. Proof. Let u = [(u ϵ ) ϵ ] G aa so u ϵ B aa ϵ I. If U is a primitive of u and U G aa then U G L because G aa G L. Conversely if U = [(U ϵ ) ϵ ] G L x then ϵ I U ϵ = u ϵ (t) dt D L so U ϵ is bounded primitive of u ϵ C aa. x 0 By the classical result of Bohl-Bohr we have U ϵ C aa consequentely U ϵ C aa D L ϵ I. By Proposition 1 U ϵ B aa ϵ I. Moreover (U ϵ ) ϵ M L i.e. k Z + m Z + U ϵ k = O ( ϵ m) ϵ 0 so (U ϵ ) ϵ M aa and U G aa. The result is independent on representatives. eferences [1] Bochner S. A new approach to almost periodicity. Proc. Nat. Acad. Sci. USA 48 (1962) 2039 2043. [2] Bochner S. Continuous mappings of almost automorphic and almost periodic functions. Proc. Nat. Acad. Sci. USA 52 (1964) 907 910. [3] Bouzar C. Khalladi M. T. Almost periodic generalized functions. Novi Sad J. Math 41 (2011) 33 42. [4] Bouzar C. Tchouar F. Z. Almost automorphic distributions. Preprint University of Oran (2014). [5] Grosser M. Kunzinger M. Oberguggenberger M. Steinbauer. Geometric theory of generalized functions. Kluwer (2001). [6] Schwartz L. Théorie des distributions. Hermann 2ième Edition 1966. [7] Veech W.A. Almost automorphic functions on groups. Amer. J. Math. 87 (1965) 719 751. ϵ eceived by the editors January 21 2015