PARTIAL DIFFERENTIAL EQUATIONS BANACH CENTER PUBLICATIONS, VOLUME 27 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 1992
AN ELEMENTARY THEORY OF HYPERFUNCTIONS AND MICROFUNCTIONS
HIKOSABUROKOMATSU
Department of Mathematics, Faculty of Science, University of Tokyo Tokyo, 113 Japan
1. Sato’s definition of hyperfunctions. The hyperfunctions are a class of generalized functions introduced by M. Sato [34], [35], [36] in 1958–60, only ten years later than Schwartz’ distributions [40]. As we will see, hyperfunctions are natural and useful, but unfortunately they are not so commonly used as distributions. One reason seems to be that the mere definition of hyperfunctions needs a lot of preparations. In the one-dimensional case his definition is elementary. Let Ω be an open set in R. Then the space (Ω) of hyperfunctions on Ω is defined to be the quotient space B (1.1) (Ω)= (V Ω)/ (V ) , B O \ O where V is an open set in C containing Ω as a closed set, and (V Ω) (resp. (V )) is the space of all holomorphic functions on V Ω (resp. VO). The\ hyper- functionO f(x) represented by F (z) (V Ω) is written\ ∈ O \ (1.2) f(x)= F (x + i0) F (x i0) − − and has the intuitive meaning of the difference of the “boundary values”of F (z) on Ω from above and below.
V Ω
Fig. 1
[233] 234 H. KOMATSU
The main properties of hyperfunctions are the following: (1) (Ω), Ω R, form a sheaf over R. B ⊂ Ω Namely, for all pairs Ω1 Ω of open sets the restriction mappings ̺Ω1 : (Ω) (Ω ) are defined, and⊂ for any open covering Ω = Ω they satisfy the B →B 1 α following conditions: S (S.1) If f (Ω) satisfies f = 0 for all α, then f = 0; ∈B |Ωα
(S.2) If fα (Ωα) satisfy fα Ωα∩Ωβ = fβ Ωα∩Ωβ for all Ωα Ωβ = , then there is∈ Ban f (Ω) such| that f = f| . ∩ 6 ∅ ∈B α |Ωα (2) The sheaf of hyperfunctions is flabby. B Ω That is, the restriction mappings ̺Ω1 are always surjective. By property (S.1) each f (Ω) has the maximal open subset of Ω on which it vanishes. Its complement is∈B called the support of f and is denoted by suppf. If F is a closed set in Ω, we write (1.3) (Ω)= f (Ω) ; supp f F . BF { ∈B ⊂ } (3) If K is a compact set in Ω, then (1.4) (Ω)= (K)′, BK A where the right hand side denotes the space of all continuous linear functionals on the locally convex space (1.5) (K) = lim (V ) −→ A V ⊃K O of all germs of real-analytic functions defined on a neighborhood of K. The pairing is given by the integral (1.6) ϕ, [F ] = ϕ(z)F (z) dz , h i −H Γ where ϕ (K), F (V K) and Γ is a closed curve in the intersection of the domain of∈Aϕ and V ∈K O, which\ encircles K once. These three properties\ characterize the sheaf of hyperfunctions. In the one- dimensional case they are derived from the K¨otheB duality [26] (1.7) (K)′ = (V K)/ (V ) , A ∼ O \ O and the Mittag-Leffler theorem (1.8) H1(V, ) = 0 for any open set V in C. O The hyperfunctions in the higher dimensional case have the same properties, and are characterized by them, too. It was not easy, however, to define the spaces (Ω), Ω Rn, of hyperfunctions having these properties. Sato had spent two yearsB before⊂ he succeeded in giving a definition for the higher dimensional case. His definition is (1.9) (Ω)= Hn(V, V Ω; ), B \ O HYPERFUNCTIONS AND MICROFUNCTIONS 235 where the right hand side is the nth relative cohomology group of the open pair (V, V Ω) with coefficients in the sheaf of holomorphic functions, which he invented\ for this purpose. The same conceptO was independently introduced by Grothendieck [12] under the name of the local cohomology group with support in Ω. In Grothendieck’s notation it is written Hn (V, ). Ω O In the earliest foundation of the theory of hyperfunctions, Martineau [30] and Harvey [13] (cf. Komatsu [22]) derived the three properties from the Martineau duality 0, p = n, (1.10) Hp (V, )= 6 K O (K)′, p = n, A for any compact set K in Rn included in an open set V in Cn, the Malgrange theorem [28] (1.11) Hn(V, ) = 0 for any open set V in Cn O and the Grauert theorem [10] saying that for any open set Ω in Rn its pseudocon- vex open neighborhoods V in Cn form a fundamental system of neighborhoods. The disadvantage of this definition is that one has to take a one-year course of several complex variables and homological algebra before he understands the fundamental concepts.
2. Hyperfunctions as boundary values of harmonic functions. As we remarked, the hyperfunctions are natural and useful. One evidence is that they are closed under taking non-characteristic boundary values of solutions of linear partial differential equations. Namely, let α (2.1) P (x, ∂)= aα(x)∂ |αX|≤m be a partial differential operator with real-analytic coefficients aα(x) on an open set V in Rn+1, and let Ω = V Rn be a non-characteristic hypersurface, or a (x′, 0) = 0. Then we have∩ (0,...,0,m) 6 (2.2) H1 (V, P ) = (Ω)m Ω B ∼ B (Komatsu [21], Schapira [39] and Komatsu–Kawai [25]). If P has constant coeffi- cients, then (2.3) H1 (V, P ) = P (V Ω)/ P (V ). Ω B ∼ B \ B Here P denotes the sheaf of hyperfunction solutions u of P u = 0. The isomor- phismsB (2.2) and (2.3) mean that the right hand side of (2.3) is identified with j ′ j ′ the m-tuples of hyperfunctions (∂nu(x , +0) ∂nu(x , 0))0≤j
If a hyperfunction f(x) (Ω) is represented by a harmonic function H(x,t) (V ), we write ∈ B ∈ P (2.6) f(x)= H(x, +0) . The basic theorems we need are the following two. The Grothendieck duality theorem ([11], [29]). Let K be a compact set in an open set V in Rn+1. Then (2.7) (K)′ = (V K)/ (V ). P ∼ P \ P If Φ (K) and H (V K), then the pairing of Φ and the class [H] of H is given∈ by P the integral ∈ P \ ∂H ∂Φ (2.8) Φ, [H] = Φ H dS, h i R ∂n − ∂n ∂L where L is a compact neighborhood with smooth boundary of K in the intersection of V and the domain of Φ.
The Mittag-Leffler theorem for harmonic functions. Let Vλ, λ Λ, be open sets in Rn+1. If a family of harmonic functions H (V V ) satisfy∈ λµ ∈ P λ ∩ µ (2.9) H + H + H = 0 on V V V λµ µν νλ λ ∩ µ ∩ ν for all V V V = , then there are harmonic functions H (V ) such that λ ∩ µ ∩ ν 6 ∅ λ ∈ P λ (2.10) H = H H on V V . λµ µ − λ λ ∩ µ In terms of cohomology groups this is equivalent to the statement (2.11) H1(V, ) = 0 for any open set V in Rn+1. P Malgrange [27] proved the theorem as the exactness of (2.12) 0 (V ) (V ) ∆ (V ) 0 , → P →E →E → where denotes the sheaf of C∞ functions (cf. H¨ormander [14], pp. 12–14). E As in the case of one complex variable, the Mittag-Leffler theorem is proved by the Runge approximation theorem saying that for a compact set K in an open set V in Rn+1, (V ) is dense in (K) if and only if V K has no relatively compact componentsP in V . The RungeP theorem can in turn\ be derived from the Grothendieck theorem. Proposition 1. (Ω) does not depend on V . B Proof. Suppose that V W are two open sets such that V and W are open. We have to prove that the natural⊂ mapping e f j : (W )/ (W ) (V )/ (V ) P P0 → P P0 induced from restriction mappings isf an isomorphism. e If H (W ) has a continuous boundary value 0, then it can be continued to ∈ P a function in (W ) by the reflection principle. Hence j is injective. P0 f 238 H. KOMATSU
The surjectivity is proved if every H (V ) is decomposed as H H ∈ P 1 − 0 with H1 (W ) and H0 0(V ). Since V = W (V W ), it follows from the Mittag-Leffler∈ P theorem∈ for P two open sets that there∩ ∪ are G (W ) and e e 1 ∈ P G0 (V W ) such that H = G1 G0. Then H0 = G0 + Gˇ0 0(V ) and H =∈G P +∪Gˇ (W ) satisfy the condition,− where ∈ P 1 1 e 0 ∈ P e Gˇ (x,t)= G (x, t). 0 − 0 − n Definition 2. Let Ω1 Ω be open sets in R . Then we choose open sets Rn+1 ⊂ V1 V in + such that V1 = V1 Ω1 V 1 and V = V Ω V are open, and define⊂ the restriction mapping ̺Ω :∪ (Ω∪) (Ω ) by the∪ natural∪ mapping e Ω1 B →B 1 e (2.13) (V )/ (V ) (V )/ (V ) P P0 → P 1 P0 1 induced from restriction mappings. e e Ω It follows from Proposition 1 that the mapping ̺Ω1 does not depend on V and V1. Theorem 1. The hyperfunctions (Ω), Ω Rn, with the restriction mappings Ω B ⊂ ̺Ω1 , form a sheaf.
P r o o f. In order to prove properties (S.1) and (S.2), let Ωλ and Vλ be open sets in Rn and Rn+1 respectively such that V = V Ω V are open in Rn+1. + λ λ ∪ λ ∪ λ We set Ω = Ω and V = V . Then V = V Ω V is also open in Rn+1. λ λ e ∪ ∪ To proveS (S.1) suppose thatS H (V ) belongs to (V ) for all λ. Then H ∈ P e P0 λ belongs to (V ) by the reflection principle. 0 e To proveP (S.2) suppose that H (V ) satisfy e λ ∈ P λ H := H H (V V ) . λµ µ − λ ∈ P0 λ ∩ µ Then it follows from the Mittag-Leffler theoreme and thee antisymmetrization that there are Fλ 0(Vλ) such that Hλµ = Fµ Fλ. Then H = Hλ Fλ is a harmonic function in ∈P(V ) which represents H (x, +0)− on Ω . − P e λ λ Theorem 2. The sheaf of hyperfunctions over Rn is flabby. B Rn+1 P r o o f. It is enough to take V = + for the representation (2.5). Definition 3. Let K be a compact set in Rn. We define the pairing of ϕ n ∈ (K) and f K (R ) as follows. Let Φ(x,t) 0(K) be the solution of the ACauchy problem∈ B ∈ P ∆Φ(x,t) = 0 , (2.14) Φ(x, 0) = 0 , ∂tΦ(x, 0) = ϕ(x) . The Cauchy–Kowalevsky theorem guarantees the existence of a unique solution. We take an H (V K) such that f(x)= H(x, +0) and define ∈ P0 \ ∂Φ ∂H (2.15) e ϕ, f := H Φ dS, h i R ∂n − ∂n S HYPERFUNCTIONS AND MICROFUNCTIONS 239 where S is a hypersurface in the intersection of V and the domain of Φ, with boundary in Ω K, and oriented as a deformation of the natural orientation of Ω in Rn, and n is\ the upward unit normal on S. Theorem 3. Under the pairing defined above we have the isomorphism (2.16) = ( (K))′ . BK ∼ A Proof. Let S be the mirror image of S in (2.15). Then S+S is the boundary − of a compact neighborhood L of K in the intersection of V and the domain of Φ. Since Φ and H are odd functions in t, the right hand side of (2.15) is equal to e the same integral over S if n is interpreted as the outer unit normal on ∂L. The upward normal n in (2.15) is also the outer unit normal on ∂L but the orientation of S is opposite to that of ∂L. Therefore the integral (2.15) should be regarded as the integral on the chain S. Hence it is equal to one half of the integral on ∂L. It is proved in the same− way that ∂Φ ∂H H Φ dS = 0 R ∂n − ∂n ∂L for any even harmonic function Φ. Hence Grothendieck’s duality theorem implies the isomorphism (2.16). When an analytic functional T ( (K))′ is given, the corresponding hyper- function f(x) is represented by the∈PoissonA integral (2.17) H(x,t) := P (x y,t), T , h − yi where 2 2 t 1 −t|ξ| +ixξ (2.18) P (x,t) := 2 2 (n+1)/2 = n R e dξ ωn+1 (x + t ) (2π) Rn is the Poisson kernel. In case the dimension n> 1, Poisson integrals H(x,t) often play an essential role in making detailed study of various classes of (generalized) functions f(x) (e.g. [43]). In this respect our definition of hyperfunctions is in accordance with the tradition of real analysis.
3. Partial differential equations with constant coefficients. Since this is a seminar on partial differential equations, we will show here some applica- tions to linear partial differential equations with constant coefficients, which are modifications, according to our new approach, of the results we reported at the Conference on Generalized Functions held in Katowice in 1966 (cf. [20], [22]). We define derivatives of hyperfunctions by α α (3.1) ∂x (H(x, +0)) = (∂x H)(x, +0) Given an r r system of differential operators 1 × 0 P (∂) : r0 r1 , B →B 240 H. KOMATSU the transposed matrix P ′( ∂) may be regarded as the homomorphism C[∂]r1 C[∂]r0 of polynomial modules− in ∂. Then by the Hilbert syzygy theorem there→ is an exact sequence terminating for some m n: ′ ′ ≤ P (−∂) P (−∂) (3.2) C[∂]r0 C[∂]r1 1 C[∂]r2 . . . ←− ←− ←− ′ Pm−1(−∂) . . . C[∂]rm−1 C[∂]rm 0 . ←− ←− ←− Theorem 4. Let P (∂) and Pj (∂) be as above. Then for any convex open set Ω in Rn we have the exact sequence
P (∂) P1(∂) Pm−1(∂) (3.3) (Ω)r0 (Ω)r1 . . . (Ω)rm 0 . B −→ B −→ −→ B −→ In particular, the equation (3.4) P (∂)u = f has a solution u (Ω)r0 if and only if the data f (Ω)r1 satisfies the com- patibility condition∈ B ∈ B
(3.5) P1(∂)f = 0 . Rn+1 Proof. Let V be a convex open set in + such that V =V Ω V is convex and open in Rn+1. Ehrenpreis [7], [8], Malgrange and H¨ormander∪ [14]∪ have proved e the theorem with replaced by the sheaf of infinitely differentiable functions. Hence the last twoB rows of the following diagramE are exact: 0 0
↓P (∂ ) ↓ P1(∂ ) (V )r0 x (V ) x . . . P −→ P −→
↓P (∂ ) ↓ P1(∂ ) (3.6) (V ) x (V ) x . . . E −→ E −→ ∆ ∆ ↓ P (∂ ) ↓ P1(∂ ) (V ) x (V ) x . . . E −→ E −→ 0↓ 0↓ By the Malgrange theorem (2.12) all columns are exact. Thus it follows that the first row is exact. By the antisymmetrization we have the exact sequence
P (∂ ) P1(∂ ) (3.7) (V )r0 x (V )r1 x . . . P0 −→ P0 −→ Hence the desired exactnesse follows from thee exact sequences 0 (V )rj (V )rj (Ω)rj 0 . → P0 −→ P −→ B → Similarly we have the followinge theorem by Harvey [13] and Bengel [1]. Theorem 5. If P (∂) is elliptic, then every hyperfunction solution u of P u = f is real-analytic wherever so is f. HYPERFUNCTIONS AND MICROFUNCTIONS 241
In particular, we have the flabby resolutions d n d d n (3.8) 0 C (1) . . . (n) 0 , −→ −→ B −→ B −→ −→ B −→ ∂ n ∂ ∂ n (3.9) 0 (1) . . . (n) 0 −→ O −→ B −→ B −→ −→ B −→ of the constant sheaf C and the sheaf of holomorphic functions over Rn and Cn respectively. O Now it is very easy to prove the basic results employed in the first foundation of the theory of hyperfunctions. Corollary 1 (Malgrange). We have (3.10) Hn(U, C) = 0 for any open set U in Rn , (3.11) Hn(V, ) = 0 for any open set V in Cn . O P r o o f. By definition we have n n Hn(U, C)= (U)(n)/d (U)(n−1). n B B n Given an n-form f (U)(n), we can extend it to a form f (Rn)(n) by ∈ B ∈ B n the flabbiness. Theorem 4 asserts the existence of a solution u (Rn)(n−1) of e∈ B du = f. Its restriction u to V satisfies du = f on V . (3.11) is proved similarly. e Corollary 2 (K¨othe–Martineau). If K is a compact set in an open set V in e e Cn such that (3.12) Hp(K, ) = 0 for p> 0 , O then we have 0, p = n, (3.13) Hp (V, ) = 6 K O ∼ (K)′, p = n. O p P r o o f. The local cohomology groups HK (V, ) are by definition the coho- mology groups of the complex O
∂ n ∂ ∂ n (3.14) 0 (Ω) (Ω)( 1) . . . (Ω)(n) 0 , −→ BK −→ BK −→ −→ BK −→ which is dual to the complex
n −∂ n −∂ −∂ −∂ (3.15) 0 (K)(n) (K)(n−1) . . . (K) 0 . ←− A ←− A ←− ←− A ←− By (3.12), the complex (3.15) is exact except for the 0-th position at which the cohomology group is equal to (K). Hence we have (3.13) by the duality of cohomology groups. O Similarly we have the following Corollary 3 (Alexander–Pontryagin). For any compact set K in an open set V in Rn we have p C n−p C ′ (3.16) HK(V, ) ∼= (H (K, )) . Hence we have a purely analytical proof of the Jordan–Brouwer theorem. 242 H. KOMATSU
4. Microlocal analysis. We have so far defined hyperfunctions as boundary values of harmonic functions and have given some applications. All of these could have been done in 1966 before the Conference in Katowice. Actually Bengel [1] had considered boundary values of solutions of elliptic equations to prove his Theorem 5. Sato had also mentioned a definition of hyperfunctions as boundary values of harmonic functions. It was three years later that microlocal analysis was born, that is, the analysis ∗ ∗ ∗ on the cosphere bundle S Ω = (T Ω 0 )/R+ or the cotangent bundle T Ω over the domain Ω on which we consider\{ solutions.} Whenever there is an epochmak- ing discovery, there are always forerunners. In this case they are Calder´on (1958) and Mizohata (1959) for singular integral operators, Kohn–Nirenberg (1965) and H¨ormander (1965) for pseudo-differential operators, and Egorov (1969) and oth- ers. The real history started, however, in 1969 at the International Conference on Functional Analysis and Related Topics held in Tokyo. Sato [37] introduced microfunctions as decompositions to the cosphere bundle S∗Ω of singularities of hyperfunctions modulo the real-analytic functions; that is, he defined a sheaf over S∗Ω such that C sp (4.1) 0 π 0 −→ A −→ B −→ ∗C −→ ∗ is exact, where π : S Ω Ω is the canonical projection and π∗ denotes the → −1 C direct image so that we have (π Ω1) ∼= (Ω1)/ (Ω1) for any open set Ω1 in Ω. C B A His motivation was to prove Theorem 5 in the variable coefficient case in a natural way. He showed that a linear differential operator P with real-analytic coefficients acts on locally in S∗Ω, that is, microlocally, and that P is injective at non-characteristicC points in S∗Ω. Hence we have (4.2) SS u SS(P u) Char P , ⊂ ∪ where SS u = suppsp u. If P is elliptic, i.e., Char P = , then u is real-analytic wherever so is P u. ∅ H¨ormander also attended the Conference and conducted a private seminar in which he talked about his idea of Fourier integral operators. His motivation was in Egorov’s work on propagation of singularities of solutions of hyperbolic equations. In 1969–1972 there was a hot competition between Sato’s school and H¨orman- der’s. The outcome was monumental papers H¨ormander [15] and Duistermaat– H¨ormander [6] on the one hand, and Sato–Kawai–Kashiwara [38] on the other. Each one is 200 pages long or more and contains many results. The highlight of both papers was that they formulated the Huygens principle mathematically and proved it. However, their definitions of fundamental concepts and their proofs of the essentially same results are completely different. Sato et al. employ the cosphere bundle S∗Ω, microfunctions , singularity spectrum SS u and quan- tized contact transforms whereas H¨ormanderC et al. the cotangent bundle T ∗Ω, HYPERFUNCTIONS AND MICROFUNCTIONS 243 wave front sets W F (u) and Fourier integral operators. The analytic wave front set W FA(u) H¨ormander introduced later (cf. [16]) coincides with the singularity spectrum SS u for distributions u but the proof was extremely difficult. One rea- son is that Sato et al. employ higher cohomology groups. For example, the sheaf of microfunctions over S∗Rn is defined as the nth derived cohomology group C n −1 a (4.3) := ∗Rn (π ) C HS O with support in S∗Rn over the comonoidal transformation (Cn Rn) S∗Rn with a non-Hausdorff topology. Our motivation is to build a bridge spanning\ ∪ these two schools.
5. Microfunctions as singularities of holomorphic functions. Let Ω be an open set in Rn. We define our disc bundle DΩ and cosphere bundle S∗Ω by (5.1) DΩ := x + iy Cn ; x Ω, y < 1 , { ∈ ∈ | | } (5.2) S∗Ω := (∂DΩ)a = x iω ; x Ω, ω = 1 , { − ∈ | | } where a stands for the antipodal mapping. Note that these are entirely different from Sato’s. Definition 4. The sheaf of microfunctions over Rn is defined by the exact sequence of sheaves over Cn: C a (5.3) 0 Rn Rn 0 , → O|[D ] → O|D →C → n n ∗ n a n where [DR ] denotes the closure DR (S R ) of DR , and A the sheaf which assigns to each open set W in Cn the∪ space O| (W A) = lim (V ) −→ O ∩ V ⊃W ∩A O for a locally closed set A. a means that we evaluate it at the antipodal point. Since the quotient sheaf Ca has the support in the closed set (S∗Rn)a, it is naturally regarded as a sheafC over (S∗Rn)a. A microfunction on a neighborhood of a point (x,ω) S∗Rn is by the defini- tion a class of holomorphic functions K(z) defined on V D∈ Rn for a neighborhood V of x iω in Cn, modulo the holomorphic functions defined∩ on a neighborhood of x iω− in Cn. Let− Σ be an open set in S∗Rn. If W is an open set in DRn such that W Σa has a fundamental system of open neighborhoods W in Cn which is pseudoconvex∪ and contains Σ as a closed set, then we have the global representation f (5.4) (Σ)= (W )/ (W Σa) . C O O ∪ In fact, we have the exact sequence of cohomology groups 0 (W Σa) (W ) (Σ) H1(W Σa, ) = 0 . → O ∪ → O →C → ∪ O Since DRn is strictly pseudoconvex and polynomially convex in Cn, we can always find such an open set W . Actually the proof of the following theorem shows that we can always take DRn for W . 244 H. KOMATSU
Theorem 6 (Kashiwara). The sheaf of microfunctions is flabby. C Proof. Let Σ be an open set in S∗Rn. It is sufficient to prove that for any open neighborhood V of DRn Σa in Cn there is a pseudoconvex open neighborhood W included in V . ∪ W = DRn is strictly pseudoconvex in the sense that it is defined in Cn by the f inequality n (5.5) ϕ(z) := y 2 = y2 < 1 | | j Xj=1 and the Hermitean form n 2 ∂ ϕ 1 2 (5.6) wkwk = w ∂zj ∂ zk 2 | | j,kX=1 is strictly positive definite. Such a function is called strictly plurisubharmonic. ∗ n a We take a partition of unity χν (z)=1on V (S R Σ) and an increasing a \a \ sequence of compact sets Kj in PΣ which covers Σ . Let ψj (z) be the sum of all χ (z) such that supp χ K = . We take an ε > 0 so small that ν ν ∩ j 6 ∅ 1 α β 1 M1 := ε1 sup ∂ ∂ ψ1(z) 2 , |α|+|β|≤2 | |≤ 4n z∈V and set M−1 = M0 = M1. If ε1,...,εj−1 have been chosen, then we set α β Mj := max εk sup ∂ ∂ ψk(z) 1≤k
Proof. If H(v) is continuous up to the boundary, then it is represented as the Poisson integral R2 v2 H(b) (5.15) H(v)= − R 2 (n+1)/2 dSb . ωn+1R ((b v) ) |b|=R −
Hence it is continued analytically to the connected component of BR in u Cn+1 ; (b u)2 = 0 for all b ∂B . The lemma will be proved if we { ∈ − 6 ∈ R} show that the component coincides with the Lie ball BR. First we note that the boundary of B is represented as R f iθ n+1 (5.16) ∂BR = e (q + ir) ; θ R, p,qf R , q + r = R, q, r = 0 { ∈ ∈ | | | | h i } (Siciak [41], Lemmaf 2). n+1 2 Secondly, we prove that if u ∂BR and b R satisfy (u b) = 0, then b R. ∈ ∈ − | |≤ f In view of (5.16) the real part v of u and the imaginary part w are written v = q cos θ r sin θ , w = q sin θ + r cos θ , − with q, r Rn+1 satisfying q + r = R and q, r = 0. ∈ | | | | h i The condition (u b)2 = 0 implies − 2 q, b cos θ 2 r, b sin θ = (q2 r2)(cos2 θ sin2 θ)+ b2 , h i − h i − − q, b sin θ + r, b cos θ = (q2 r2) cos θ sin θ . h i h i − 246 H. KOMATSU
Hence we have, in view of q + r = R, | | | | 2 q, b = cos θ 2 q R + (b2 R2) , 2 r, b = sin θ 2 r R + (b2 R2) . h i { | | − } h i − { | | − } If either q =0 or r =0, then it is easy to see that b = R. Otherwise, we have the following components of b with respect to the orthogonal| | unit vectors q/ q and r/ r : | | | | q b2 R2 r b2 R2 , b = cos θ R + − , , b = sin θ R + − . q 2 q r − 2 r | | | | | | | | Adding the squares of both sides, we have R b2 R2 R b2 R2 (b2 R2) cos2 θ 1 − + sin2 θ 1 − 0. − − q − 4q2 − r − 4r2 ≥ | | | | Since R b2 R2 ( q r )2 b2 1 − = | | − | | − , − q − 4q2 4q2 | | we have cos2 θ sin2 θ (b2 R2) ( q r )2 b2 + 0 . − { | | − | | − } 4q2 4r2 ≥ This implies ( q r )2 b2 R2. | | − | | ≤ ≤ The proof also shows that for any u ∂BR there is a b ∂BR satisfying (u b)2 = 0. Hence the Lie ball B is a component∈ of u Cn∈+1 ; (b u)2 = 0 R f for− any b ∂B . { ∈ − 6 ∈ R} f The following lemma is a part of the Paley–Wiener theorem for hyperfunctions. n n Lemma 3. Let K be a compact convex set in R . If f(x) K (R ), then its Fourier transform ∈B (5.17) f(ζ) := R e−ihx,ζif(x) dx is an entire function on Cnbsatisfying the estimates (5.18) f(ζ) C exp H (ζ)+ ε ζ | |≤ ε { K | |} for any ε> 0 with a constant Cε depending on ε, where
(5.19) HK(ζ)= sup Im x,ζ . x∈K h i Moreover, the Poisson integral of f is represented as
1 −t|ξ|+ihx+iy,ξi (5.20) Pf(x + iy,t)= n R e f(ξ) dξ , (2π) Rn b and hence is holomorphic on the tube domain x + iy Cn ; y
It is easy to prove that the Fourier transformation : K ExpK and Rn+1 F B → the integral I : ExpK 0( K) defined by (5.20) are continuous linear mappings. If f(x)=δ(x→y P), y K,\ then (I )f is clearly equal to the Poisson − ∈ ◦ F integral Pf. The linear combinations of such δ(x y) are dense in K and the n+1 − B Poisson integral P : K 0(R K) is continuous. Hence we have the identity (5.20) for all f B. → P \ ∈BK Definition 5. Let f (Ω) be a hyperfunction on an open set Ω in Rn. We ∈B Rn+1 choose an upper neighborhood V + of Ω so large that every closed ball of Rn+1 ⊂ radius 1 in + touching an x Ω is entirely in V except for x, and we take a harmonic function H (V ) such∈ that f(x) = H(x, +0). Then it follows from Lemmas 1 and 2 that ∈H( Px, 1) has an analytic continuation H(x + iy, 1) to π−1Ω and its class in a(π−1Ω) does not depend on the choice of the defining harmonic function H(x,t),C which we define to be sp f:
(5.21) (sp f)(x,ω) := [H(x + iy, 1)]y=−ω .
V
(x, 1)
Ω x Fig. 2 Theorem 7. The spectral mapping sp induces the isomorphism (5.22) (Ω)/ (Ω) ∼ (π−1Ω) B A →C for any open set Ω in Rn. P r o o f. As we remarked after Lemma 1 this is equivalent to the exactness of the sequence (4.1) of sheaves. Therefore it is sufficient to prove (5.22) for any relatively compact open set Ω in Rn. To prove the surjectivity, let [K(x + iy)] be a class in (DRn)/ (DRn ∗ a −1 O O ∪ S Ω ) ∼= (π Ω). We have to find an H(x,t) (V ), for a sufficiently large upper neighborhoodC V of Ω, such that ∈ P (5.23) K(x + iy) H(x + iy, 1) (S∗Ωa) . − ∈ O n We choose a relatively compact smooth domain Ω0 in R such that all balls of radius √2 with center in Ω are included in Ω0, and define for t> 1 (5.24) H(x,t) := P (x w,t 1)K(w) dw , R − − Ω0 248 H. KOMATSU where P (x,t) is the Poisson kernel. This is originally a harmonic function on the upper half space t > 1 and has an analytic continuation to (x + iy,t) ; t > 1, x Rn, y
Ω0
Ωs(ω)
x Fig. 3
If f(x) is a real-analytic function on a compact domain Ω0, then the same proof shows that its Poisson integral
H(x,t) := P (x y,t)f(t) dy R − Ω0 has an analytic continuation H(x + iy, 1) (DΩ S∗Ωa) for the interior Ω ∈ O ∪ of Ω0. Rn+1 Finally, suppose that an H(x,t) ( + ) has the analytic continuation H(x + iy, 1) (DΩ S∗Ωa) for a relatively∈ P compact domain Ω. Then we can ∈ O ∪ HYPERFUNCTIONS AND MICROFUNCTIONS 249
construct as above a harmonic function K(x,t) on Ω t 0 such that × { ≥ } K(x + iy, 1) = H(x + iy, 1) , x + iy DΩ . ∈ Hence it follows from the reflection principle that H(x,t)= K(x,t)+ K(x, 2 t) H(x, 2 t) . − − − The right hand side being real-analytic up to t = 0, H(x, +0) is real-analytic on Ω.
6. Boundary values of holomorphic functions on tuboids. Let Ω be an open set in Rn, and Γ a convex open cone in Rn 0 . An open set W in Cn is said to be a tuboid of profile Ω + iΓ if W Ω + iΓ\{, and} if for any open subset n−1⊂ n−1 Ω0 ⋐ Ω and subcone Γ0 ⋐ Γ (i.e. Γ0 S ⋐ Γ S ) there is an r> 0 such that ∩ ∩ Ω + i y Γ ; y < r W . 0 { ∈ 0 | | }⊂ For any holomorphic function F (z) (W ) we define its boundary value F (x + iΓ 0) (Ω) as follows. ∈ O ∈B Let Ω and Γ be as above. We take a small γ Γ and define 0 0 ∈ 0 (6.1) H (z,t) := P (z w,t)F (w) dw . γ R − Ω0+iγ This is originally a harmonic function on (x+iy,t) ; t> 0, x Rn, y γ
iΓ0
Ω0 + iγ
Ωs
Ωγ Fig. 4 Definition 6. For a holomorphic function F (x + iy) (W ) on a tuboid of ∈ O profile Ω + iΓ , its boundary value on Ωγ is defined by (6.2) F (x + iΓ 0) := H (x, +0) , x Ω . γ ∈ γ 250 H. KOMATSU
It is easily proved that F (x + iΓ 0) does not depend on γ and Ω0 on the common domain of definition. Hence the boundary value F (x + iΓ 0) is actually defined on Ω. We write F (z) (Ω + iΓ 0) if it is a holomorphic function on a tuboid W of profile Ω +iΓ . Thus∈ Othe boundary value is a linear mapping (Ω +iΓ 0) (Ω). O →B Theorem 8. Let Γ be a convex open cone in Rn 0 . A hyperfunction f(x) (Ω) on an open set Ω in Rn has its singularity spectrum\{ } ∈ B (6.3) suppsp f Ω (Γ ◦ Sn−1) ⊂ × ∩ if and only if it is the boundary value F (x + iΓ 0) of a holomorphic function F (z) on a tuboid of profile Ω + iΓ . Here Γ ◦ denotes the polar of Γ : (6.4) Γ ◦ := ξ Rn ; y,ξ 0 for all y Γ . { ∈ h i≥ ∈ } The correspondence between F (z) (Ω + iΓ 0) and F (x + iΓ 0) (Ω) is one- to-one. If F (x + iΓ 0) vanishes on∈ an O open subset, it vanishes on∈B its connected component. Proof. Suppose that F (z) (Ω + iΓ 0). Then H (x + iy,t) constructed ∈ O γ above is holomorphic on Ω0 +i γ +y ; y
Therefore, our microfunctions are isomorphic to Sato’s. The support suppsp f as a microfunction is called the singularity spectrum of the hyperfunction f, and is denoted by SS f. The following theorem is called Martineau’s edge of the wedge theorem [31]. Theorem 9. Let Γ , j = 1,...,m, be convex open cones in Rn 0 , and set j \{ } (6.5) Γjk := Γj + Γk. Then we have for any open set Ω in Rn the isomorphism m m (6.6) f (Ω);SS f Ω Γ ◦ Sn−1 = (Ω + iΓ 0)/ , ∈B ⊂ × j ∩ ∼ O j ∼ n j[=1 o Mj=1 where (Fj (z) (Ω + iΓj 0)) 0 if and only if there are Fjk(z) (Ω + Γjk0) such that ∈ O ∼ ∈ O
(6.7) Fjk(z)+ Fkj(z) = 0 , m (6.8) Fj(z)= Fjk(z) . Xk=1 n Here Γjk are convex open cones but they are not necessarily in R 0 . In that case (Ω + iΓ 0) denotes the real-analytic functions on Ω. \{ } O jk P r o o f. We consider the linear mapping
bd : (Ω + iΓj 0) (Ω) M O →B which sends (Fj (z)) into Fj (x + iΓj 0). Its image is clearly in the left hand side of (6.6). If f(x) (PΩ) has its singularity spectrum in Ω Γ ◦, then by ∈ B × j the flabbiness of the microfunctions, sp f is decomposed into theS sum sp fj ◦ with SS fj Ω Γj . Each fj is written the boundary value Fj (x + iΓj0)P of an F (z) (⊂Ω +×iΓ 0). Since f f is real-analytic, it may be added to one j ∈ O j − j of fj. P Suppose that F (z) (Ω + iΓ 0) satisfy m F (x + iΓ 0) = 0 in (Ω). j ∈ O j j=1 j j B If m = 1, then we have F1(z) 0 by TheoremP 8. In general, SS Fm(x + iΓm0) ≡ is included in Ω (Γ ◦ m−1 Γ ◦). Since Γ ◦ = Γ ◦ Γ ◦ cover Γ ◦ m−1 Γ ◦, × m ∩ j=1 j mj m ∩ j m ∩ j=1 j there are F (z) (Ω+SiΓ 0) such that F (z)= m−1 F (z). LetSF (z)= mj ∈ O mj m j=1 mj jm Fmj(z) and subtract it from Fj (z). Then we have theP same situation with m 1 −functions. − The converse is trivial. ◦ Rn If Γj covers , we have a representation of all hyperfunctions on Ω as the sum{ of} the boundary values of holomorphic functions on tuboids of profile Ω + iΓj0. Kaneko [17] adopted this as the definition of hyperfunctions.
7. Hyperfunctions and microfunctions with a prescribed singularity or regularity. Applications. We have so far considered hyperfunctions and 252 H. KOMATSU associated microfunctions. There are, however, infinitely many classes of gener- alized functions as well as regular functions between the hyperfunctions and the real-analytic functions. An advantage of our approach to hyperfunctions and mi- crofunctions is that it is then very easy to characterize those classes of generalized functions or regular functions among the hyperfunctions and hence to introduce the corresponding classes of microfunctions. For the sake of brevity we consider here only Schwartz’ distributions ′, the ultradistributions (s)′ and {s}′ of Gevrey classes and the associatedD classes , (s) and {s} ofD regular functions.D We denote by the empty symbol or (s) or Es Efor 1 Cα, with a constant Cα if = , α ∗ ∅ (7.1) sup ∂ f(x) Ch|α| α !s, for any h> 0 with a constant C if = (s), x∈K | |≤ Ch|α||α|!s, with constants h and C if = s . ∗ | | ∗ { } Clearly ∗(Ω), Ω Rn, form a sheaf. ∗(Ω) denotes its subclass of all func- tions with compactE support⊂ in Ω. The spaceD ∗(Ω) has a natural locally convex topology and its dual ∗′(Ω) is defined to beD the space of (ultra-)distributions D Ω ∗′ ∗′ of class . If Ω1 Ω, then the restriction mapping ̺Ω1 : (Ω) (Ω1) is defined naturally,∗ ⊂ and ∗′(Ω), Ω Rn, form a sheaf. We haveD natural→ D inclusion relations: D ⊂ (7.2) (s) {s} (t) ′ (t)′ {s}′ (s)′ , A⊂E ⊂E ⊂E ⊂E⊂D ⊂ D ⊂ D ⊂ D ⊂B 1 An (ultra-)differential operator P (∂) of class is defined by ∗ ∞ α (7.6) P (∂)= aα∂ , |αX|=0 where a C, and the a vanish for sufficiently large α if = ; α ∈ α | | ∗ ∅ (7.7) a Ck|α|/( α !)s | α|≤ | | with constants k and C if = (s); and (7.7) holds for any k > 0 with a constant C if = s . ∗ ∗ { } Theorem 11. Under the same assumptions on Ω and V the following condi- tions are equivalent for H(x,t) (V ): ∈ P (a) H(x, +0) ∗(Ω); ∈E (b) H(x,t) converges in ∗(Ω) as t 0; E → (c) For any differential operator P (∂) of class , P (∂x)H(x,t) is locally bounded in a neighborhood of Ω. ∗ Under each of these conditions, H(x,t) converges to H(x, +0) in ∗(Ω). E A proof of Theorems 10 and 11 is given in [24]. Similarly we have the following Theorem 12. The following conditions are equivalent for a holomorphic func- tion F (z) on a tuboid of profile Ω + iΓ : (a) The boundary value F (x + iΓ 0) in the sense of hyperfunctions belongs to ∗′(Ω) (resp. ∗(Ω)); D E (b) F (x + iy) converges in ∗′(Ω) (resp. ∗(Ω)) as y tends to zero in some D E (and any) closed cone Γ1 ⋐ Γ ; (c) For any compact set K in Ω and closed cone Γ1 ⋐ Γ there are a growth function G(t) of class and constants C and r such that ∗ (7.8) sup F (x + iy) C exp G( y ) , y Γ1 , y < r x∈K | |≤ | | ∈ | | (resp. P (∂z)F (z) is bounded on K + i y Γ1 ; y < r for any ultradifferential operator P (∂) of class ). { ∈ | | } ∗ Under each of these conditions, the topological limit coincides with F (x+iΓ 0). From now on we write = ( ) instead of = to avoid confusion. ∗ ∞ ∗ ∅ ∗ n Definition 7. We denote by DRn (resp. ∗ DRn ) the sheaf over C of holomorphic functions on DRn withO the| growth conditionO | of class (resp. with the regularity condition of class ), and define the sheaf ∗ of microfunctions∗ of class (resp. the sheaf ∗ of microfunctions∗ of regular classC ) by the exact sequence∗ C ∗ ∗ ∗ a 0 Rn Rn ( ) 0 → O|[D ] → O |D → C → (7.9) a (resp. 0 Rn Rn ( ) 0). → O|[D ] → O∗|D → C∗ → 254 H. KOMATSU Then we have the exact sequences ∗′ sp ∗ (7.10) 0 π∗ 0 , →A→D sp→ C → (7.11) 0 ∗ π 0 . →A→E → ∗C∗ → ∗ The sheaves and ∗ are not flabby but have a little weaker property of suppleness. HenceC we haveC Martineau’s edge of the wedge theorem for ∗′ and ∗. D E (∞) ′ The microfunctions and (∞) corresponding to the distributions and the infinitely differentiableC functionsC have been introduced by Bony [3]D and Bengel–Schapira [2]. 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