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REAL ANALYTIC GENERALIZED FUNCTIONS 1. Introduction And

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REAL ANALYTIC GENERALIZED FUNCTIONS 1. Introduction And REAL ANALYTIC GENERALIZED FUNCTIONS S. PILIPOVIC,´ D. SCARPALEZOS, V. VALMORIN Abstract. Real-analytic generalized functions are investigated and the analytic singular support and analytic wave front of a generalized func- tion in G(Ω) are introduced and described. 2000 Mathematics Subject Classification. 46F10, 46S10, 35 A27. Keywords and phrases. Algebra of generalized functions. 1. Introduction and definitions ¯ Holomorphic generalized functions are defined as the solutions to @zi G = 0; i = 1; :::; p; where G is an element of a generalized function algebra G(Ω); Ω ½ Cp (see [3], [4], [5], [11], [19] and [24] for the definition and the properties for generalized function algebra). Recall, elements of this algebra are classes of equivalences of nets of smooth functions with respect to an ideal N (Ω) which consists of nets (n")" converging to zero, on com- pact sets, faster than any positive power of " as " ! 0: So, @G¯ = 0 means a @G¯ " = O(" ) on compact sets of Ω, for any a > 0. Thus, various properties for such generalized functions have became interesting and important. The p algebra GH (Ω); Ω ½ C ; of holomorphic generalized functions is introduced and studied in [4], [1], [6] and [7] while in [22] is proved the existence of a global holomorphic representative for an f 2 GH (Ω); Ω ½ C: Moreover, in [16] is shown (in the one dimensional case) that a holomorphic generalized function equals zero if it is equal to zero at every classical point. In this paper we are interested in the class of real analytic generalized d functions GA(!);! is open in R ; in order to study generalized analytic microlocal regularity of generalized functions. We give in Section 2 a representation of an arbitrary generalized function f 2 G(!), ! is open in Rd; as a boundary value on I ½½ ! of a corresponding d holomorphic generalized function [(F")"] on I £ (¡1=´; 1=´) which means ¯ d (@zi F")" 2 N (I £ (¡1=´; 1=´) ); i = 1; :::; d. Further, in the case d = 1 we give a global real analytic representative of a real analytic generalized function. Then we define analytic singular support of a generalized function. Also in Section 2 we prove that a generalized real analytic function equals zero if it is equal to zero at every classical point of a subset of ! of positive measure by proving the same result for a generalized holomorphic function in an open set of Cp (see [16] for the case p = 1). Generalized analytic wave front of f 2 G(!) is introduced in Section 3 (see [14], [8], [9], [10] and [15] for the local and microlocal G1-properties of Colombeau-type generalized 1 2 S. PILIPOVIC,´ D. SCARPALEZOS, V. VALMORIN functions). It is proved, for a distribution, that its wave front coincides with the generalized wave front of the embedded distribution in G(!): Actually we prove a stronger result, the strong associativity of a distribution and a generalized function is necessary and sufficient condition for the equality of their analytic wave fronts. Section 4 is devoted to the comparison of the notions of S-analyticity in the sense of [2] and the analyticity of this paper. 1.1. Definitions. We recall the main definitions ([5], [11], [19], [20]). Let ! be an open set in Rd and E(!) be the space of smooth functions with (®) the sequence of seminorms ¹º(Á) = supfjÁ (x)j; ® · º; x 2 Kºg; º 2 N0, where (Kº)º is an increasing sequence of compact sets exhausting !: Then the set of moderate nets EM (!), respectively of null nets N (!); consists of (0;1) nets (f")" 2 E(!) with the properties a (8n 2 N)(9a 2 R)(¹n(f") = O(" )); b respectively, (8n 2 N)(8b 2 R)(¹n(f") = O(" )) (O is the Landau symbol ”big O”). Both spaces are algebras and the latter a is an ideal of the former. Putting vn(r") = supfa; ¹n(r") = O(" )g and en((r")"; (s")") = exp(¡vn(r" ¡ s")); n 2 N; we obtain (en)n, a sequence of ultra-pseudometrics on EM (!) defining the ultra-metric topology (sharp topology) on G(!): The simplified Colombeau algebra G(!) is defined as the quotient G(!) = EM (!)=N (!): This is also a differential algebra. If the nets (f")" consist of constant functions on ! (i.e. the seminorms ¹n reduce to the absolute value), then one obtains the corresponding spaces of nets of complex (or real) numbers EM and N0: They are algebras, N0 is an ideal in EM and, as a quotient, one obtains the Colombeau algebra of generalized complex numbers C¯ = EM =N0 (or R¯). It is a ring, not a field. The sharp topology in C¯ is defined as above. Note, a ball B˜(x0; r), where x0 = [(x0;")"] 2 C¯ and r > 0; in the sharp topology is given by ¡ ln r B˜(x0; r) = f[(x")"]; jx0;" ¡ x"j = O(" )g: (0;1) Recall ([11]), !M is the set of nets (x")" 2 ! with the property jx"j = O("¡a) for some a > 0 and the relation of equivalence is introduced by a (x")" » (y")" iff jx" ¡ y"j = O(" ) for every a > 0: The corresponding classes of equivalence determine! ˜, and! ˜c denotes its subset consisting of the classes of equivalence with the compactly supported representatives; (x")" is compactly supported by K ½½ ! if x" 2 K for " < "0. The ultrametric in! ˜ defined in the same way as in the one dimensional case for C¯. Note C˜ = C¯ (R˜ = R¯). The embedding of Schwartz distributions in E0(!) is realized through the 0 sheaf homomorphism E (!) 3 f 7! [(f ¤ Á"j!)"] 2 G(!); where the fixed net ¡d d of mollifiers (Á")" is defined by Á" = " Á(¢="); " < 1; where Á 2 S(R ) satisfies Z Z m n Á(t)dt = 1; t Á(t)dt = 0; m 2 N0 ; jmj > 0: REAL ANALYTIC GENERALIZED FUNCTIONS 3 m m1 mn 0 (t = t1 :::tn and jmj = m1 + ::: + mn). E (!) is embedded into Gc(Ω) of compactly supported generalized functions and the sheaf homomorphism, extended onto D0, gives the embedding of D0(!) into G(!): Generalized algebra Gc(!) consists of elements in G(!) which are com- 1 1 pactly supported while G (!) is defined in [19] as the quotient EM (!)=N (!); 1 (0;1) where EM (!) consists of nets (f")" 2 E(!) with the property d (n) a (8K ½½ !)(9a 2 R)(8n 2 N0)(j sup f" (x)j = O(" )); x2K Note that G1 is a subsheaf of G: If Ω is an open set of Cp ´ R2p then a generalized function f 2 G(Ω) is a ¯ holomorphic generalized function, if @zi f = 0, i = 1; :::; p; ([4]) i.e. for any representative (f")" of f, 1=2(@xi + i@yi )f"(x; y) 2 N (Ω), i = 1; :::; p. In the ¯ ¯ sequel, we will write @f = 0 for @zi f = 0, i = 1; :::; p: It is proved in [4] that for every open bounded set of W ½½ Ω there exists a representative (f")" of f in W such that f" is holomorphic in W for every " < 1: By the sheaf property of G, one obtains that Ω !GH (Ω) is a sheaf. It is a subsheaf of G1: 2. Real analytic generalized functions d Definition 1. Let ! denote an open set in R and x0 be a point of !.A function f 2 G(!) is said to be real analytic at x0 if there exist an open ball B = B(x0; r) ½ ! containing x0 and (g")" 2 EM (B) such that (i) fjB = [(g")"] in G(B); (ii)(9´ > 0)(9a > 0)(9"0 2 (0; 1)) (®) j®j+1 ¡a d sup jg" (x)j · ´ ®!" ; 0 < " < "0; ® 2 N : x2B It is said that f is real analytic in ! if f is real analytic at each point of !. The space of all generalized functions which are real analytic in ! is denoted by GA(!). The analytic singular support, singsupp gaf; is the complement of the set of points x 2 ! where f is real analytic. It follows from the definition that GA is a subsheaf of G. Using Stirling’s formula it is seen that condition (ii) in Definition 1 is equivalent to (iii)(9´ > 0)(9a > 0)(9"0 2 (0; 1)) (®) j®j ¡a d sup jg" (x0)j · ´(´j®j) " ; 0 < " < "0; ® 2 N0: x2B The use of Taylor expansion and condition (ii) of Definition 1 imply that d g"; " < "0 admit holomorphic extensions in a complex ball B = fz 2 C ; jz ¡ x0j < rg which is independent of ". Consequently we get a holomorphic d d extension G of [(g")"] and then fjB \ R = GjB \ R . 4 S. PILIPOVIC,´ D. SCARPALEZOS, V. VALMORIN The following theorem gives a representative of a real analytic generalized function in a neighborhood of a point. We will use a multidimensional notation: ® ®1 ®d d ®! = ®1!:::®d!; ´ = ´ :::´ p; ´ > 0; ® = (®1; :::; ®d) 2 N ; ; j j1 j d d (iy) = (iy1) :::(iyd) d ; i = ¡1; y 2 R ; j = (j1; :::; jd) 2 N ; In the one dimensional case (with ³ = » + i´; »; ´ 2 R), ¯ @f(³) = 1=2(@» ¡ i@´)f(»; ´), @f(³) = 1=2(@» + i@´)f(»; ´): Theorem 1. Let f 2 G(!) and I ½½ ! be an open d¡interval, I = I1 £ ::: £ Id. Let σ denote a positive moderate function defined in (0; 1] (σ(") · σ(") p(1="); " < 1 for some polynomial p) such that lim"!0 ln " = ¡1. Then there exists ´ > 0 and "0 2 (0; 1) such that a net (F")" defined for (x; y) 2 d I £ (¡1=´; 1=´) and " 2 (0;"0) by X (j) j F"(x; y) = f" (x)(iy) =j! j1·σ(");:::;jd·σ(") d represents a holomorphic generalized function F = [(F")"] in I£(¡1=´; 1=´) such that f = F (¢; 0) in I (and we say that f is the boundary value of F ).
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