On the singular spectrum of tempered ultrahyperfunctions corresponding to proper convex cones

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1 Rend. Sem. Mat. Univ. Padova 1xx (201x) Rendiconti del Seminario Matematico della Università di Padova c European Mathematical Society On the singular spectrum of tempered ultrahyperfunctions corresponding to proper convex cones Daniel H.T. Franco - Magno B. Alves Abstract We address in this paper the issue of singularities of ultrahyperfunctions. Following the Carmichael s approach for ultrahyperfunctions, we study the relation between the singular spectrum of a class of tempered ultrahyperfunctions corresponding to proper convex cones and their expressions as boundary values of holomorphic functions. In passing, a simple version of the celebrated edge of the wedge theorem for this setting is derived from the integral representation without using cohomology. Mathematics Subject Classification (2010). 46F20, 81T05. Keywords. Singular spectrum, tempered ultrahyperfunctions. 1. Introduction In 1958 Sebastião e Silva [28, 29] introduced the class of generalized functions called by him tempered ultradistributions, which are the Fourier image of L. Schwartz distributions of exponential type. Later, Hasumi [16] considered the global theory of tempered ultrahyperfunctions in the higher dimensional space, and Morimoto [19, 20, 21] (who coined the name tempered ultrahyperfunctions) localized the theory of tempered ultrahyperfunctions in the imaginary direction. The interest in ultrahyperfunctions arose simultaneously with the growing interest in various classes of analytic functionals and various attempts to develop a theory of such functionals which would be analogous to the Schwartz theory of distributions. Since then, various developments of the theory have been proposed by many authors, among others we refer the reader to [35, 22, 8, 9, 10, 34, 26, 30, 3, 27, 15, 14, 4, 13, 5, 6, 25, 7, 12]. More recently, the study of ultrahyperfunctions has been stimulated by relativistic quantum field theory with a fundamental length [3, 14, 4, 5]. Using the cohomological approach, Morimoto gave a clear description of the cotangential components of singularities of ultrahyperfuntions in Ref. [22] (the ultrahyperfunctions were called in [22] cohomological ultradistributions), not necessarily assumed to be tempered. Following Nishiwada [23, 24], we present an alternative way of describing the singularities of a class of tempered ultrahyperfunctions corresponding to proper convex cones in terms of generalized boundary values of holomorphic functions; namely, the singular spectrum of a tempered ultrahyperfunction u is characterized by the directions from which the boundary values can be taken in an analytic representation of u. Our approach for tempered ultrahyperfunctions parallels that of Carmichael addressed in [8, 9, 10]. In this approach, boundary values of holomorphic functions in tube domains with edge an open set X of are tempered ultrahyperfunctions in X; and conversely a tempered ultrahyperfunction in is the sum of boundary values of holomorphic functions in tube domains. Thus, an analysis of the arguments of Nishiwada and those of Carmichael shows that the connection between the singular spectrum and the boundary values of holomorphic functions can be expressed naturally in the setting of ultrahyperfunctions. We are therefore in the ideal situation to study the singularities of tempered Daniel H.T. Franco, Universidade Federal de Viçosa, Departamento de Física, Av. Peter Henry Rolfs s/n, Campus Universitário, Viçosa, MG, Brasil, CEP: daniel.franco@ufv.br Magno B. Alves, Universidade Federal de Juiz de Fora, Departamento de Matemática, Campus Universitário, Bairro Martelos, Juiz de Fora, MG, Brasil, CEP: magno branco@yahoo.com.br

2 2 Daniel H.T. Franco - Magno B. Alves ultrahyperfunctions in a manner which follows hyperfunction theory as closely as possible. An important tool for the analysis leading to this study is the extension of the Paley-Wiener-Schwartz theorem for this class of generalized functions. In addition, we prove the edge of the wedge theorem for tempered ultrahyperfunctions corresponding to a convex cone. 2. Tempered ultrahyperfunctions in a nutshell To begin with, we shall recall very briefly the basic definition of tempered ultrahyperfunctions. Firstly, we shall consider the function h K (ξ) = sup ξ, x, ξ, x K the indicator of K, where K is a compact set in. h K (ξ) < for every ξ since K is bounded. For sets K = [ k, k ] n, 0 < k <, the indicator function hk (ξ) can be easily determined: h K (ξ) = sup ξ, x = k ξ, ξ, ξ = x K n ξ i. Let K be a convex compact subset of, then H b ( ; K) (b stands for bounded) defines the space of all functions ϕ C ( ) such that e hk(ξ) D α ϕ(ξ) is bounded in for any multi-index α. One defines in H b ( ; K) seminorms (1) ϕ K,N = sup ξ ;α N i=1 { e h K (ξ) D α ϕ(ξ) } <, N = 0, 1, 2,.... The space H b ( ; K) equipped with the topology given by the seminorms (1) is a Fréchet space [16, 19]. If K 1 K 2 are two compact convex sets, then h K1 (ξ) h K2 (ξ), and thus the canonical injection H b ( ; K 2 ) H b ( ; K 1 ) is continuous. Let O be a convex open set of. To define the topology of H( ; O) it suffices to let K range over an increasing sequence of convex compact subsets K 1, K 2,... contained in O such that for each i = 1, 2,..., K i Ki+1 (K i+1 denotes the interior of K i+1) and O = i=1 K i. Then the space H( ; O) is the projective limit of the spaces H b ( ; K) according to restriction mappings above, i.e. (2) H( ; O) = lim proj K O H b ( ; K), where K runs through the convex compact sets contained in O. By H ( ; O) we denote the dual space of H( ; O). Proposition 2.1 (Hasumi [16, Proposition 3], Morimoto [19, Theorem 5]). A distribution V H ( ; O) may be expressed as a finite order derivative of a continuous function of exponential growth where g(ξ) is a bounded continuous function. V = D γ ξ [eh K(ξ) g(ξ)], In the space C n of n complex variables z i = x i + iy i, 1 i n, we denote by T () = + i C n the tubular set of all points z, such that y i = Im z i belongs to the domain, i.e., is a connected open set in called the basis of the tube T (). Let K be a convex compact subset of, then H b (T (K)) defines the space of all continuous functions ϕ on T (K) which are holomorphic in the interior T (K ) of T (K) such that the estimate (3) ϕ(z) M K,N (ϕ)(1 + z ) N is valid. The best possible constants in (3) are given by a family of seminorms in H b (T (K)) { } (4) ϕ K,N = inf M K,N (ϕ) sup z T (K) (1 + z ) N ϕ(z) <, N = 0, 1, 2,.... If K 1 K 2 are two convex compact sets, we have that the canonical injection (5) H b (T (K 2 )) H b (T (K 1 )),

3 On the singular spectrum of tempered ultrahyperfunctions corresponding to proper convex cones 3 is continuous. Given that the spaces H b (T (K i )) are Fréchet spaces, with topology defined by the seminorms (4), the space H(T (O)) is characterized as a projective limit of Fréchet spaces: (6) H(T (O)) = lim proj K O H b (T (K)), where K runs through the convex compact sets contained in O and the projective limit is taken following the restriction mappings above. Let K be a convex compact set in. Then the space H(T (K)) is characterized as a inductive limit (7) H(T (K)) = lim ind K 1 K H b(t (K 1 )), where K 1 runs through the convex compact sets such that K is contained in the interior of K 1 and the inductive limit is taken following the restriction mappings (5). The Fourier transformation is well defined on the space H( ; O). If ϕ H( ; O), the Fourier transform of ϕ belongs to the space H(T (O)), for any open convex nonempty set O. By the dual Fourier transform H ( ; O) is topologically isomorphic with the space H (T ( O)) [19]. Remark 2.2. We will put H = H(C n ) = H(T ( )) and, as usual, we shall denote the dual space of H by H. H. Definition 2.3. A tempered ultrahyperfunction is by definition a continuous linear functional on In this article we are interested in the class of tempered ultrahyperfunctions corresponding to proper convex cones. Therefore, we start by recalling some terminology and simple facts concerning cones. An open set C is called a cone if C (unless specified otherwise, all cones will have their vertices at zero) is invariant under positive homoteties, i.e., if for all λ > 0, λc C. A cone C is an open connected cone if C is an open connected set. Moreover, C is called convex if C +C C and proper if it contains no any straight line (observe that if C is a proper cone, it follows that if y C and y 0 then y / C). A cone C is called compact in C we write C C if the projection prc def = C S n 1 prc def = C S n 1, where S n 1 is the unit sphere in. Being given a cone C in y-space, we associate with C a closed convex cone C in ξ-space which is the set C = { ξ ξ, y 0, y C }. The cone C is called the dual cone of C. Definition 2.4. Let C be a proper open convex cone, and let C C. Let B[0; r] denote a closed ball of the origin in of radius r, where r is an arbitrary positive real number. We define by { T (C \(C B[0;r])) = x + iy C n x, y ( C \ ( C B[0; r] )) }, y < <, where > 0 is an arbitrary but fixed number, the truncated tube domain. We are going to introduce a space of holomorphic functions which satisfy certain estimate according to Carmichael [8, 9, 10]. We want to consider the space consisting of holomorphic functions f such that (8) f(z) M(C )(1 + z ) N e h C (y), z = x + iy T (C \(C B[0;r])), where h C (y) = sup { ξ, y ξ C } is the supporting function of C, M(C ) is a constant that depends on an arbitrary compact cone C and N is a non-negative real number. Clearly, as C is a convex set, one has h C (y) = h ch C (y), where ch C is the convex hull of C. The set of all functions f which for every cone C C are holomorphic in T (C \(C B[0;r])) and satisfy the estimate (8) will be denoted by Hc o. Remark 2.5. Here, it should be noted that the truncated cone acts such as to ensure that the supporting function h C (y) is finite. In fact, we have ξ, y ξ y < ξ, and since C is a cone, if we take αξ, with α > 0, instead of ξ, then our old supremum is multiplied by α to get the new supremum, but this new supremum must be still finite if we define = α in order to obtain a new upper bound for y.

4 4 Daniel H.T. Franco - Magno B. Alves Remark 2.6. The space of functions Hc o constitutes a generalization of the space Ai of Sebastião ω e Silva [28] and the space A ω of Hasumi [16] to arbitrary tubular radial domains in C n (a tube domain is said to be radial if its base is a connected cone in ). We now shall introduce another space of holomorphic functions whose elements are analytic in a domain which is larger than T (C \(C B[0;r])) and has boundary values in. Let B(0; r) denote an open ball of the origin in of radius r, where r is an arbitrary{ positive real number. Let T (C \(C B(0;r))) denote the subset of C n defined by T (C \(C B(0;r))) = x + iy C n x, y ( C \ ( C B(0; r) )), y < < }, where > 0 is an arbitrary but fixed number. Definition 2.7. Let C be a proper open convex cone, and let C C. Denote by T (C ) the tube domain { } x + iy C n x, y C, y < < We say that the function f = f(z) is in the space Hc o if it is holomorphic in T (C ) and satisfies the estimate (9) f(z) M(C )(1 + z ) N e h C (y), z = x + iy T (C \(C B(0;r))). Note that Hc o Hc o for any open convex cone C. Following Hasumi [16], we define the kernel of the mapping f : H C by Π, where Π is the set of all pseudo-polynomials in one of the variables z 1,..., z n. We recall that a pseudo-polynomial is a function the form α zα j G α(z 1,..., z j 1, z j+1,..., z n ), where G α (z 1,..., z j 1, z j+1,..., z n ) are functions in Hc o with respect to (z 1,..., z j 1, z j+1,..., z n ). Then, f Hc o belongs to the kernel Π if and only if f(z), ψ(x) = 0, with ψ H and x = Re z. Put U c = Hc o /Π, that is, U c is the quotient space of Hc o by set of pseudo-polynomials Π. Definition 2.8. The set U c is the subspace of the tempered ultrahyperfunctions generated by H o c corresponding to a proper open convex cone C. Definition 2.9. We denote by H C (Rn ; O) the subspace of H ( ; O) of distributions of exponential growth with support in the cone C : { (10) H C (Rn ; O) = V H ( ; O) supp(v ) C }. 3. Boundary values of holomorphic functions In this section the boundary values of holomorphic functions are always considered in the distribution sense defined below. We say that f T (C ) has a boundary value U = BV (f) in H as y 0, y C C, if for all ψ H the limit U, ψ = lim f(x + iy)ψ(x) dx, y 0 y C exists. The following theorem shows that functions in Hc o have distributional boundary values in H (T (O)). Further, it shows that functions in Hc o satisfy a strong boundedness property in H (T (O)). Theorem 3.1 (Franco [13, Theorem 5.2]). Let C be a proper open convex cone, and let C C. Let V = D γ ξ [eh K(ξ) g(ξ)], where g(ξ) is a bounded continuous function on and h K (ξ) = k ξ for a convex compact set K = [ k, k ] n. Let V H C ( ; O). Then (i) f(z) = (2π) n V, e i ξ,z is an element of Hc o, { (ii) f(z) y = Im z C C, y } is a strongly bounded set in H (T (O)), where is an arbitrarily but fixed positive real number, (iii) f(z) F 1 [V ] H (T (O)) in the strong (and weak ) topology of H (T (O)) as y = Im z 0, y C C.

5 On the singular spectrum of tempered ultrahyperfunctions corresponding to proper convex cones 5 The functions f Hc o can be recovered as the (inverse) Fourier-Laplace transform of the constructed distribution V H C (Rn ; O). This result is a version of the Paley-Wiener-Schwartz theorem in the tempered ultrahyperfunction set-up. Theorem 3.2 (Franco [13, Theorem 5.3]). Let f Hc o, where C is a proper open convex cone. Then the distribution V H C (Rn ; O) has a uniquely determined inverse Fourier-Laplace transform f(z) = (2π) n V, e i ξ,z which is holomorphic in T (C ) and satisfies the estimate (8), with B[0; r] replaced by B(0; r). Remark 3.3. We would like to emphasize that in Theorems 4.1 and 4.2 above, we are considering the inverse Fourier-Laplace transform f(z) = (2π) n V, e i ξ,z, in contrast to the inverse Fourier-Laplace transform f(z) = (2π) n V, e i ξ,z adopted in Theorems 5.2 and 5.3 in [13]. This is due to the convention of signs in the Fourier transform which is used here. The same proof as in Carmichael [9, Theorem 1, equation (4)] combined with the proofs of Theorems 3.1 and 3.2 shows that the following proposition is true. Proposition 3.4. Let C be a proper open convex cone, and let C C. Let f Hc o. Then there exists a unique element V H C (Rn ; O) such that (11) f(z) = F 1[ e ξ,y V ], z T (C \(C B(0;r))), where (11) holds as an equality in H (T (O)). At this point we must clarify a fact that may not have been clear in Refs. [14, 13]. Let H C (T (O)) denote the subset of H (T (O)) defined by H C (T (O)) = { U H (T (O)) U = F 1 [V ], V H C (Rn ; O) }. Then, the boundary value mapping given in (iii), Theorem 3.1, maps Hc o on H C (T (O)). This follows immediately because of the recovery of the elements f Hc o as the (inverse) Fourier-Laplace transform of the constructed distribution V H C (Rn ; O) and the fact that the Fourier and inverse Fourier transforms from H C (T (O)) to H C (Rn ; O) and H C (Rn ; O) to H C (T (O)), respectively, are isomorphisms! In fact, it is easy to verify the Proposition 3.5. U c is isomorphic to the space H C (T (O)), with the isomorphism being defined by the mapping BV. If u U c and U H C (T (O)) is the corresponding element under the isomorphism, then (12) BV C (f), ψ = lim f(x + iy)ψ(x) dx = U, ψ, ψ H, y 0 y C where f H o c is any representative of u. Remark 3.6. We use the notation BV C (f) in order to emphazise that the boundary value of f is obtained taking the limit y = Im z 0 from the directions of the cone C. Corollary 3.7 (See Sebastião e Silva [28, Proposition 11.1]). Every element u U c is represented by a holomorphic function f Hc o under the form (13) f(z) = V (ξ)e i ξ,z dξ, C where V H C (Rn ; O). We will note it by u = [f]. Remark 3.8. In what follows, we always shall understand the results below as related to the subset H C (T (O)) of H (T (O)). Our next step is to prove that the mapping f BV (f) is injective. We do this in a constructive way, beginning by proving the following Theorem 3.9. Let C n be an open set. Then, H(T ( )) is dense in H().

6 6 Daniel H.T. Franco - Magno B. Alves An indication of proof of this theorem was given in [19]. Here, our approach to this problem makes use of a construction of a suitable approximation of the identity modeled on the kernel of the heat equation due to Baouendi-Treves [1, 32]. This construction has been a key tool for many function and distributional spaces [18] (see also [2]). Proof of Theorem 3.9. To begin with, let B 0, B 1 be two open balls, both centered at the origin, and Φ = (Φ 1,..., Φ n ) : B 0 B 1 a C map. Assume a map Z : B 0, where } (14) = {Z(x) = x + iy x B 0, y = Φ(x) B 1 C n, is an open set. Furthermore, Φ j (0) = 0 and Φ j (0)/ = 0, j = 1,..., n. Note that ( ) ( Zj = jk + i Φ ) j (15), 1 j,k n 1 j,k n are elements of a nonsingular complex square matrix. We will assume that (16) and we may assume that Φ(x) 1 < x, 2 x B0, (17) Φ j < 1 2, x B. 0 Take, for τ > 0, the following Gaussian function in C n B 0 : E τ (z; x) = (τ/π) n/2 e τ z Z(x) 2, where the notation z Z(x) 2 means n ( zj Z j (x) ) 2. Then, given any holomorphic function ϕ H(), we define the basic operator in the Baouendi-Treves approximation formula: ϕ τ (z) def = [K τ ϕ](z) = It is clear that ϕ τ is an entire function of z. Lemma ϕ τ H(T ( )). B 0 ( ) Zj E τ (z; x)ϕ(z(x)) det dx. Proof of Lemma It is convenient to make the hypothesis that Z j C (B 0 ), j = 1,..., n. Let Q k = { z C n Im z k; j = 1,..., n }. Thus, ( ) sup (1 + z ) N ϕ τ (z) sup (1 + z ) N E τ (z; x)ϕ(z(x)) det Zj dx z Q k z Q k B 0 M sup (1 + z ) N E τ (z; x) z Q k B 0 (1 + x ) M ( ) det Zj dx. Taking k diam B 0 (possibly after decreasing the radius of B 0 ), we obtain that E τ (z; x) (τ/π) n/2 e τ Re z x 2 e τk2. ( ) Zj We also observe that det is bounded because of (16). Hence, sup (1 + z ) N ϕ τ (z) M (τ/π) n/2 e τk 2 z Q k sup Re z (1 + k + Re z ) B0 N 2 τ Re z x (1 + x ) M e dx. Finally, note that (1 + k + Re z ) N 2 τ Re z x (1 + x ) M e, is a bounded function of Re z and x for M > N. In turn, this estimate can be continued by sup (1 + z ) N ϕ τ (z) M (τ/π) n/2 e τk 2 z Q k This proves that ϕ τ (z) H(T ( )). 2 τ Re z x e dx <.

7 On the singular spectrum of tempered ultrahyperfunctions corresponding to proper convex cones 7 The following lemma can easily be established using standard techniques. Lemma Let ϕ τ be a bounded sequence in H() and ϕ H(). If the sequence of functions ϕ τ converges to ϕ uniformly, as τ, then ϕ τ converges to ϕ in the topology of H() defined by semi-norms ϕ,n = sup z (1 + z ) N ϕ(z). From now on we fix our attention on the convergence of ϕ τ ϕ. At this point, we use an argument used in [2, page 55]: modifying the functions Φ j off neighborhood of B 0 may assume without loss of generality that the functions Φ j (x), j = 1,..., n are defined throughout, have compact support and satisfy (17) everywhere, that is (18) Φ j < 1 2, x Rn, and ϕ τ (z) = ( ) Zj E τ (z; x)ϕ(z(x)) det dx. Now, consider the following restriction of the entire functions ϕ τ to : ( ) ϕ τ (Z(x )) = (τ/π) n/2 e τ Z(x ) Z(x) 2 Zj (19) ϕ(z(x)) det dx, Introducing the change of variables x x + τ 1/2 x in the integral (19), we get ϕ τ (Z(x )) = π n/2 e τ Z(x ) Z(x +τ 1/2 x) (20) 2 Γ(Z(x + τ 1/2 x)) dx, where we define Γ(Z(x + τ 1/2 x)) def. = ϕ(z(x + τ 1/2 x)) det ( Zj x k ) (x + τ 1/2 x). It is worth emphasizing that the differentiability of Z at x allows us to write Z(x + τ 1/2 x) = Z(x ) + Z x (x ) τ 1/2 x + R(τ 1/2 x), where Z x denotes the n n matrix (15) and R(τ 1/2 x) is the remainder term. We also note that R(τ 1/2 x) τ 1/2 x τ + 0, uniformly for x B 0. We have then that the exponential in the integral (20) rewrites as with e Z x (x )x 2 e (τ 1/2 R(τ 1/2 x)) ψ(τ,x,x), ψ(τ, x, x) def. = 2Z x (x )x + τ 1/2 R(τ 1/2 x). Note that ψ(τ, x, x) is uniformly bounded for τ > 0 and x, x B 0. Lemma 3.12 (Berhanu-Cordaro-Hounie [2, Lemma II.1.3]). Let B be an n n matrix with real coefficients and norm B < 1 and A = I + i B, where I is the identity matrix. Then (det A)π n/2 e Ax 2 dx = 1. Let x be an arbitrary but fixed point of B 0. Since x is fixed, the matrix (15) satisfies the hypotheses of Lemma 3.12 in view of (18). Hence, for a holomorphic function ϕ H(), we may write ϕ(z(x )) = π n/2 )x 2 Γ(Z(x )) dx, e Z x (x

8 8 Daniel H.T. Franco - Magno B. Alves where (again) Z x in the exponential denotes the n n matrix (15) and ( ) Γ(Z(x )) def. = ϕ(z(x Zj )) det (x ). x k Then, ϕ τ (Z(x )) ϕ(z(x )) = π n/2 e Z x (x )x 2{ e (τ 1/2 R(τ 1/2 x)) ψ(τ,x,x) } Γ(Z(x + τ 1/2 x)) Γ(Z(x )) dx. In order to estimate this difference, we first note that the differentiability of ϕ(z(x + τ 1/2 x)) and det ( Z j / x k ) (x + τ 1/2 x) implies that Γ(Z(x + τ 1/2 x)) = Γ(Z(x )) + τ 1/2 Γ (Z(x )) Z x (x )x + S(τ 1/2, x, x), with uniformly for x, x B 0. Hence, S(τ 1/2,x,x) τ 1/2 τ + 0, ϕ τ (Z(x )) ϕ(z(x )) = I 1 + I 2, where and I 1 = π n/2 )x 2{ e (τ 1/2 R(τ 1/2 x)) ψ(τ,x,x) 1} Γ(Z(x )) dx, e Z x (x I 2 = π n/2 τ 1/2 )x 2{ e (τ 1/2 R(τ 1/2 x)) ψ(τ,x,x) Γ (Z(x ))Z x (x )x} dx. e Z x (x To estimate I 2 we observe that e Z x (x )x 2 = e x 2 + Φ x (x )x 2 e (3/4) x 2, in view of (18). Moreover, it follows from (3) that ϕ is bounded for x, while det ( Z j / x k ) (x ) is bounded in B 0. Thus, I 2 admits the following estimate uniform for x, x B 0 : I 2 M 1 τ 1/2 e (3/4) x 2 x dx M 2 τ 1/2, where M 2 is a positive constant. In turn, to estimate I 1 we first observe that e (τ 1/2 R(τ 1/2 x)) ψ(τ,x,x) 1 = [ (τ 1/2 R(τ 1/2 x)) ψ(τ, x, x) ] Θ(τ, x, x), where Θ(τ, x, x) is uniformly bounded for τ > 0 and x, x B 0. Hence, I 1 admits the following estimate I 1 M 3 τ 1/2 R(τ 1/2 x). Recalling that R(τ 1/2 x)) is the rest of differentiability, we have that τ 1/2 R(τ 1/2 x) < ε, once τ >> 0, with ε > 0 arbitrarily fixed. In summary, for each ε > 0, arbitrarily fixed, corresponds τ o such that ϕ τ (Z(x )) ϕ(z(x )) M 3 ε + M 2 τ 1/2, x B 0, if τ τ o (where the constants M 3, M 2 depend on ε and τ o ). Therefore, it follows from Lemma 3.11 that ϕ τ (Z(x )) converges to ϕ(z(x )) uniformly on, as τ, w.r.t. the semi-norm,n. This completes the proof of Theorem 3.9. Remark It should be mentioned that another proof of this theorem essentially along these lines is contained in [3, Theorem 2.13]. There, as for the proof of the theorem above, the authors have used a construction inspired by an argument of Hörmander [17, Proposition 9.1.2].

9 On the singular spectrum of tempered ultrahyperfunctions corresponding to proper convex cones 9 Finally, we take advantage of the approximation scheme described above in order to prove that the mapping f BV (f) is injective. Theorem If the boundary value BV (f) of f Hc o vanishes identically, then the function f itself vanishes. Proof. The following proof is similar to the proof of Theorem II.1.2 of Ref. [11]. It is convenient to assume that the cone C is connected (then we may assume that C is also connected). Consider an open set U (with U smooth). Define, for any ϕ H µ γ U,f, ϕ = f(z)ϕ(z) dz, ϕ H, U+iγ where the vector γ C, with γ <. The above formula defines a functional carried by U + iγ. Suppose now that BV (f) 0; this means that BV (f) U = 0 and hence µ γ U,f is carried by U. Thus, given an open neighborhood N of U in C n we can choose an ε, with 0 < ε <, and a constant M γ (C ) > 0 so that (21) f(z)ϕ(z) dz M γ(c ) sup (1 + z ) N ϕ(z). U+iγ z N for all γ C, γ < ε. Take, for τ > 0, ϕ(z) = (τ/π) n/2 e τ z x iγ 2, where z x iγ 2 = n ( ) 2, zj x j iγ j with x U away from N. Then, on the basis of approximation scheme described in Theorem 3.9, we conclude that the left-hand side of Eq.(21) converges to f(x + iγ). On the other hand, the right-hand side is dominated by M γ (C )(τ/π) n/2 sup (1 + ε + Re z ) N e τ z x 2 e τε2, z N Then, if we select N sufficiently thin around U to ensure that there is an open subset of, U 1 U, such that ε << dist(x, N ), x U 1, z N, the right-hand side will converge to zero, implying that f = 0 in an open of the type U 1 + ic. It now follows from connectedness of T (C ) that f 0, which is what we wanted to prove. 4. Carrier of analytic functionals in H (T (O)) and ultrahyperfunctional singular spectrum We address in this section the issue of singularities of ultrahyperfunctions. Following the Carmichael s approach for ultrahyperfunctions, we study the relation between the singular spectrum of the tempered ultrahyperfunctions corresponding to proper convex cones and their expressions as boundary values of holomorphic functions. Of course, it will be important to specify in which sense the boundary values are considered. This issue will be addressed in Lemma 4.3. Let be a closed set in T (O). Let m be a closed neighborhood of defined by m = { z C n dist(z, ) 1/m }. For a closed set m of C n, H b ( m ) is the space of all continuous functions ψ on m which are holomorphic in the interior of m and satisfy { ψ m,n = sup (1 + z ) N ψ(z) }. z m N N H b ( m ) is a Fréchet space with the seminorms ψ m,n. If m < m, m m, then we have the canonical injections (22) H b ( m ) H b ( m ). We define the space H() (23) H() = lim ind m H b( m ),

10 10 Daniel H.T. Franco - Magno B. Alves where the inductive limit is taken following the restriction mappings (22). Given that is a closed set in T (O), Λ T (O) general the transpose of Λ T (O), t Λ T (O) Hahn-Banach theorem that the injectivity of t Λ T (O) Λ T (O) (H(T (O))) in H(). : H(T (O)) H() defines the restriction map. In : H () H (T (O)), need not be injective. It follows from : H () H (T (O)) is equivalent to the density of Definition 4.1. An analytic functional U H (T (O)) is said to be carried by the closed set C n, if for every closed neighborhood m of the functional U belongs to the range of t Λ T (O) m. Proposition 4.2 below shows that an analytic functional U H (T (O)) is carried by a closed set C n if, and only if, there is a decreasing sequence { m } m=1 of closed neighborhoods of such that, for every m, the functional U is already a functional on the space H(T (O)) m of restrictions to m of functions in H(T (O)), where H(T (O)) m carries the topology induced by H(T (O)). Proposition 4.2. For U H (T (O)) to be carried by the closed set of T (O) it is necessary and sufficient that there be a fundamental sequence of closed neighborhood m of and a constant C > 0 such that, for all ψ H(T (O)) (24) { U, ψ C sup (1 + z ) N ψ(z) }. z m N N Proof. Suppose that there is Ũ H () such that U = t Λ T (O) Ũ. Then, given any ψ H(T (O)), U, ψ = ψ. There is a fundamental sequence of closed neighborhood m of and a constant Ũ, ΛT (O) C such that Ũ, ψ C ψ m,n, ϕ H b ( m ), for every m, whence (24). Conversely, suppose that (24) holds. First we note that Λ T (O) ψ U, ψ is a well defined linear functional on the subspace Λ T (O) (H(T (O))) of H(): by (24), Λ T (O) ψ = 0 U, ψ = 0. The right hand side of (24) defines a continuous seminorm not only on Λ T (O) the linear functional Λ T (O) that satisfies (24). Clearly U = t Λ T (O) (H(T (O))) but on H() too. By the Hahn-Banach theorem, (H(T (O))) to H() as a linear functional Ũ Ũ. ψ U, ψ extends from Λ T (O) In particular, when is contained in = { z C n z = x + iy, x, y = 0 }, every function f Hc o, which for each y C as a function of x = Re z belongs to H (T (O)), is a continuous linear functional on the space of restrictions to of functions in H(T (O)). Then, according to Theorem 3.1(iii), U = BV (f) the distributional boundary value of f is an element of H (T (O)) carried by. Let C be an open cone of the form C = m C j, m <, where each C j is an proper open convex cone. If we write C C, we mean C = m C j with C j C j. Furthermore, we define by Cj = { } ξ ξ, x 0, x C j the dual cones of Cj, such that the dual cones Cj, j = 1,..., m, have the properties (25) \ and m Cj, (26) C j C k, j k, j, k = 1,..., m, are sets of Lebesgue measure zero. Assume that V H C (Rn ; O) can be written as V = m V j, where we define (27) V j = D γ ξ [eh K(ξ) λ j (ξ)g(ξ)], with λ j (ξ) denoting the characteristic function of C j, j = 1,..., m, g(ξ) being a bounded continuous function on and h K (ξ) = k ξ for a convex compact set K = [ k, k ] n.

11 On the singular spectrum of tempered ultrahyperfunctions corresponding to proper convex cones 11 Lemma 4.3. Let C be an open cone such that C = m C j, where the C j are open convex cones such that (25) and (26) are satisfied and V H C (Rn ; O) be represented as V = m V j where V j = D γ ξ [eh K(ξ) λ j (ξ)g(ξ)], with λ j (ξ) denoting the characteristic function of Cj, j = 1,..., m, g(ξ) being a bounded continuous function on and h K (ξ) = k ξ for a convex compact set K = [ k, k ] n. Assume that F 1 [V ] H (T (O)) is carried by. Then, U = F 1 [V ] is the sum of distributional boundary values in H (T (O)) of functions f j (z) Hc o j, j = 1,..., m. Proof. Consider (28) h y (ξ) = f(z) P (iz) e i ξ,z dx, z T (C \(C B(0;r))), with h y (ξ) = e k ξ g y (ξ), where g y (ξ) is a bounded continuous function on, and P (iz) = ( i) γ z γ. By hypothesis f Hc o and satisfies (8), with B[0; r] replaced by B(0; r). For this reason, for an n-tuple γ = (γ 1,..., γ n ) of non-negative integers conveniently chosen, we obtain (29) f(z) M(C )(1 + z ) n ε e h C (y), P (iz) where n is the dimension and ε is any fixed positive real number. This implies that the function h y (ξ) exists and is a continuous function of ξ. Further, by using arguments paralleling the analysis in [33, p.225] and the Cauchy-Poincaré Theorem [33, p.198], we can show that the function h y (ξ) is independent of y = Im z. Therefore, we denote the function h y (ξ) by h(ξ). From (29) we have that f(z)/p (iz) L 2 as a function of x = Re z, y C \ ( C B(0; r) ). Hence, from (28) and the Plancherel theorem we have that e ξ,y h(ξ) L 2 as a function of ξ, and (30) f(z) P (iz) = F 1[ e ξ,y h(ξ) ] (x), z T (C \(C B(0;r))), where the inverse Fourier transform is in the L 2 sense. From Parseval-type relation for the inverse Fourier transform of a distribution V of exponential growth, we have that (31) F 1 [V ], ψ = V, F 1 [ψ] = = = = m D γ ( ξ λj (ξ)h(ξ) ) 1, (2π) n ψ(x)e i ξ,x dx m λ j (ξ)h(ξ), ( 1) γ (2π) n D γ ξ ( ψ(x)e i ξ,x ) dx m λ j (ξ)h(ξ), ( i) γ (2π) n x γ ψ(x)e i ξ,x dx m lim C y 0 F 1[ e ξ,y λ j (ξ)h(ξ) ], ( i) γ (x + iy) γ ψ(x), where we have used the fact that the differentiation under the integral sign is valid. We note that ψ(x) H(T (O)) implies (z γ ψ(x)) H(T (O)) as a function of x = Re z. From (30), we have for z T (C \(C B(0;r))) i γ (x + iy) γ f(x + iy), ψ(x) = F 1[ e ξ,y h(ξ) ], ψ(x), (32) m = F 1[ e ξ,y λ j (ξ)h(ξ) ], ψ(x).

12 12 Daniel H.T. Franco - Magno B. Alves Combining (31) and (32), we obtain m m F 1 [V ], ψ = lim f j(x + iy), ψ(x) = U j, ψ = U, ψ. C y 0 Thus the inverse Fourier transform F 1 [V ] is a distributional boundary value of m f j(z) in the sense of weak convergence. But from [31, Corollary 1, p.358] the latter implies strong convergence since H(T (O)) is Montel. We will now describe the singularities of a tempered ultrahyperfunction. We start by observing that the relationship between the distributional boundary value and the (analytic) singular spectrum of a tempered ultrahyperfunction u, which will be denoted by S.S.(u), is that if the family of functions f j (x + iy) is holomorphic in T (C \(C B(0;r))), then the singular spectrum of the distributional boundary value U = m lim C y 0 f j (x + iy) is contained in X C, where X is an open set in and C is the dual cone of C = m C j. Here we recall that, according to Proposition 3.5, the tempered ultrahyperfunction u is related to U as follows: u = m BV (f C j ). In short, by considering the Paley- Wiener-Schwartz Theorem 3.2, the singular spectrum of a tempered ultrahyperfunction u is characterized by the directions from which the boundary values can be taken in an analytic representation of u, as shown in the following Theorem 4.4. Let X be an open set in and let u U c (X), where U c (X) is the subspace of the tempered ultrahyperfunctions generated by Hc o corresponding to a proper open convex cone C. If V H C (Rn ; O) (with O ), then S.S.(u) X C. Proof. Let {C j } j L be a finite covering of closed properly convex cones of C. Decompose V H C (Rn ; O) as follows [8, Theorem 4]: (33) V = m V j, such that V j H C j (Rn ; O) = { } V j H ( ; O) supp(v j ) Cj. Next apply the Theorem 3.2 for each V j. Then, in view of the Lemma 4.3, the decomposition (33) will induce a representation of u in the form of a sum of boundary values of functions f j Hc o j, such that f j F 1 [V j ] H in the strong topology of H as y = Im z 0, y C j C j. According to Theorem 3.2, the family of functions f j satisfy the estimate fj (z) M(C )(1 + z ) N e h C j (y), z = x + iy T (C j \(C j B(0;r))). unless ξ, Y < 0 for ξ Cj and Y C j, with Y <. Then the cones of bad directions responsible for the singularities of these boundary values are contained in the dual cones of the base cones. So, we have the inclusion (34) S.S.(u) X j C j. Then, by making a refinement of the covering and shrinking it to C, we obtain the desired result. Corollary 4.5. Let condition S.S.(u) X C of Theorem 4.4 be satisfied. Then, u = m BV C (f j ), where f j H o c j. From Theorem 3.2, Proposition 3.5, Corollary 3.7 and Lemma 4.3, we can draw the following Corollary 4.6. Let C be an open convex cone and let C be an arbitrary compact cone contained in C. Let C be the dual cone of C. If an element U H is the boundary value in the distributional sense of a function f which is analytic in T (C ) and which satisfy the estimate (8), then S.S.(U) X C. In what follows, we give a version of the celebrated edge of the wedge theorem for the space of the tempered ultrahyperfunctions. It is derived from the integral representation without using cohomology (see [22, 34] for a proof using the cohomological approach). We note that in the literature one finds some global versions of the edge of the wedge theorem for tempered ultrahyperfunctions, e.g., Refs. [21, 30, 13]. In turn, in Ref. [7] the authors propose a local version of the same theorem for tempered ultrahyperfunctions.

13 On the singular spectrum of tempered ultrahyperfunctions corresponding to proper convex cones 13 Theorem 4.7. Let C j, j = 1, 2 be proper open convex cones. Assume that the distributional boundary values of two holomorphic functions f j Hc o j (j = 1, 2) agree, that is, U = BV (f 1 ) = BV (f 2 ), where U H in accordance with the Theorem 3.1. Then f 1 and f 2 can be glued together to a common element f which is holomorphic on T (ch(c)) = + i ch(c), where ch(c) is the convex hull of the cone C = C 1 C 2. Proof. Let U = BV (f 1 ) = BV (f 2 ). It is not difficult to show that C 1 C 2 = (C 1 C 2 ). Then, from Theorem 4.4 and Corollaries 4.5 and 4.6 we have S.S.(U) X ( C 1 C 2 ). If C 1 C 2 =, it follows that S.S.(U) =, which means that U is a real analytic function. On the other hand, if C 1 C 2, then there exists a holomorphic function f such that U = BV (f). Since U = BV (f) = BV (f 1 ) = BV (f 2 ), it follows that the differences f f j, (j = 1, 2), vanish on the domain of definition of each f j by Theorem This proves the theorem. References [1] M.S. Baouendi and F. Treves, A property of the functions and distributions annihilated by a locally integrable system of complex vector fields, Ann. of Math. 113 (1981) 387. [2] S. Berhanu, P.D. Cordaro and J. Hounie, An introduction to involutive structures, Cambridge University Press, [3] E. Brüning and S. Nagamachi, Relativistic quantum field theory with a fundamental length, J. Math. Phys. 45 (2004) [4] E. Brüning and S. Nagamachi, Solution of a linearized model of Heisenberg s fundamental equation II, J. Math. Phys. 49 (2008) [5] E. Brüning and S. Nagamachi, Frame independence of the fundamental length in relativistic quantum field theory, J. Math. Phys. 51 (2010) [6] E. Brüning and S. Nagamachi, Solution of a linearized model of Heisenberg s fundamental equation, in ADVANCES IN QUANTUM FIELD THEORY, Edited by Sergey Ketov, InTechWeb. org, pg.185, [7] E. Brüning and S. Nagamachi, Edge of the wedge theorem for tempered ultrahyperfunctions, Comp. Var. and Ellip. Eq. 59 (2014) 787. [8] R.D. Carmichael, Distributions of exponential growth and their Fourier transforms, Duke Math. J. 40 (1973) 765. [9] R.D. Carmichael, The tempered ultra-distributions of J. Sebastião e Silva, Portugal. Math. 36 (1977) 119. [10] R.D. Carmichael, Distributional boundary values and the tempered ultra-distributions, Rend. Sem. Mat. Univ. Padova 56 (1977) 101. [11] P.D. Cordaro and F. Treves, Hyperfunctions on hypo-analytic manifolds, Princeton University Press, [12] A. Debrouwere and J. Vindas, On the non-triviality of certain spaces of analytic functions. Hyperfunctions and ultrahyperfunctions of fast growth, Rev. R. Acad. Cienc. Exactas Fs. Nat. Ser. A, in press, doi.org/ /s , (preprint form) arxiv: v2. [13] D.H.T. Franco, Holomorphic extension theorem for tempered ultrahyperfunctions, Portugal. Math. 66 (2009) 175. [14] D.H.T. Franco, J.A. Lourenço and L.H. Renoldi, Ultrahyperfunctional approach to non-commutative quantum field theory, J. Phys. A: Math. Theor. 41 (2008) [15] D.H.T. Franco and L.H. Renoldi, A note on Fourier-Laplace transform and analytic wave front set in theory of tempered ultrahyperfunctions, J. Math. Anal. Appl. 325 (2007) 819. [16] M. Hasumi, Note on the n-dimensional ultradistributions, Tôhoku Math. J. 13 (1961) 94. [17] L. Hörmander, The analysis of linear partial differential operators I, Springer Verlag, Second Edition, [18] J. Hounie and P. Malagutti, On the convergence of the Baouendi-Treves approximation formula, Comm. in P.D.E. 23 (1998) [19] M. Morimoto, Theory of tempered ultrahyperfunctions I, Proc. Japan Acad. 51 (1975) 87. [20] M. Morimoto, Theory of tempered ultrahyperfunctions II, Proc. Japan Acad. 51 (1975) 213. [21] M. Morimoto, Convolutors for ultrahyperfunctions, Lecture Notes in Physics, vol.39, Springer-Verlag, 1975, p.49.

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