A brief proof of Bochner's tube theorem and a generalized tube
aa r X i v : . [ m a t h . C V ] J u l A brief proof of Bochner’s tube theorem andthe local version due to Kashiwara
J. Noguchi ∗ The University of Tokyo
Abstract
The aim of this note is to give a brief straightforward proof of Bochner’s tube theorem by showingdirectly that two points of the envelope of holomorphy of a tube can be connected by a line segment.We then discuss a generalization by M. Abe, and the local version due to Kashiwara. We will give anexplicit expression of the convex cones contained in the envelope of holomorphy of the local or finitetube.
Keywords: tube domain; analytic continuation; envelope of holomorphy; Oka’s boundary distancetheorem.MSC2020: 32D10, 32Q02, 32A45.
1. Tube Theorem
The following statement is well-known as Bochner’s tube theorem:
Theorem 1.1 (Bochner, Stein ( n = 2)) . Let T R be a tube (domain) of C n with real base R ⊂ R n . Thenthe envelope of holomorphy of T R is T co( R ) , where co( R ) denotes the (affine) convex hull of R . We use the following two basic theorems: As for the envelope of holomorphy we add the constructiveexistence for a convenience as an appendix (cf. § C n are always unramified . Theorem 1.2.
Every holomorphically separable domain D over C n admits an envelope of holomorphy,containing D as a subdomain. In particular, a univalent (schlicht) domain Ω of C n admits an envelopeof holomorphy (multi-sheeted in general), containing Ω as a subdomain. Theorem 1.3 (Boundary distance: Oka [16], [17] VI (1942), IX (1953); [10]; [15]) . If D / C n is a domainof holomorphy over C n , then − log δ ( ζ, ∂ D ) ( ζ ∈ D ) is a continuous plurisubharmonic function, where δ ( ζ, ∂ D ) denotes the distance function to the boundary (cf. § . A tube domain or simply a tube T R with real base R which is a real domain of R n , is defined by(1.4) T R = R + i R n = { ( z j ) ∈ C n : ( ℜ z j ) ∈ R } , where ℜ z j stands for the real part of z j .A tube T R is convex if and only if R = co( R ), and a convex domain of C n is a domain of holomorphy,so that T co( R ) is a domain of holomorphy.We first give a brief simple proof of the above tube theorem. By the same idea, we discuss a general-ization by M. Abe [1] to the case of unramified domains with giving the counter-examples if the lengthsof the tubes are cut off in finite (see § § Proof of Theorem 1.1.
The case of n = 1 is trivial. We assume n ≥ ∗ Research supported in part by Grant-in-Aid for Scientific Research (C) 19K03511. et π : ˆ T → C n be the envelope of holomorphy of T R by Theorem 1.2. With ˆ R := ˆ T ∩ π − R n , ̟ = π | ˆ R : ˆ R → R n is a real (unramified) domain over R n (i.e., ̟ is a local homeomorphism and ˆ R isconnected) and ̟ ( ˆ R ) ⊂ co( R ). Then ˆ T has a structure of a tube in the following sense:(1.5) π : ˆ T = ˆ R + i R n −→ R n + i R n = C n . It follows from Oka’s boundary distance Theorem 1.3 that − log δ ( ζ, ∂ ˆ T ) is plurisubharmonic and satisfies(1.6) − log δ ( ζ, ∂ ˆ T ) = − log δ ( ζ + iy, ∂ ˆ T ) , ∀ y ∈ R n . With the local coordinates π ( p ) = ( x j + iy j ), if δ ( p, ∂ ˆ T ) is of C -class, it satisfies the semi-positivedefiniteness:(1.7) (cid:18) ∂ ∂z j ∂ ¯ z k − log δ ( ζ, ∂ ˆ T ) (cid:19) j,k = (cid:18) ∂ ∂x j ∂x k − log δ ( ζ, ∂ ˆ T ) (cid:19) j,k ≥ . We define a line segment L [ p, q ] ⊂ ˆ R connecting two distinct points p, q ∈ ˆ R as follows. Let L [ ̟ ( p ) , ̟ ( q )] ⊂ R n be a line segment connecting ̟ ( p ) and ̟ ( q ). Then there is a unique connectedcomponent L p of the inverse ̟ − L [ ̟ ( p ) , ̟ ( q )], containing p . If L p ∋ q , we write L p = L [ p, q ] ⊂ ˆ R . Formutually close p, q ∈ ˆ R , L [ p, q ] exists, but in general the existence is unknown at this moment. Assumingthe existence of L [ p, q ], we see by (1.7) that the restricted function − log δ ( ζ, ∂ ˆ T ) | L [ p,q ] , even if it is notdifferentiable, is a convex function on the line segment L [ p, q ]. Therefore we have(1.8) min L [ p,q ] δ ( ζ, ∂ ˆ T ) = min { p,q } δ ( ζ, ∂ ˆ T ) . We put the diagonal set ∆ = { ( p, p ) ∈ ˆ R : p ∈ ˆ R } . Note that ˆ R \ ∆ is connected, since n ≥ If S := n ( p, q ) ∈ ˆ R \ ∆ : ∃ L [ p, q ] ⊂ ˆ R o ⊂ ˆ R \ ∆, then S = ˆ R \ ∆D Firstly, S is non-empty and open. It suffices to show that S is closed in ˆ R \ ∆ . Let ( p, q ) ∈ ˆ R \ ∆ be an accumulation point of S . Then there is a sequence of points ( p ν , q ν ) ∈ S ( ν = 1 , , . . . ) such thatlim ν →∞ p ν = p, lim ν →∞ q ν = q, L [ p ν , q ν ] ⊂ ˆ R. By (1.8) there is a constant ρ > ν such that the tubular neighborhood U ν (univalent) ofevery L [ p ν , q ν ] with width ρ is contained in ˆ R . Then for every sufficiently large ν , U ν ∋ p, q D Therefore L [ p, q ] ⊂ U ν ⊂ ˆ R ; thus, ( p, q ) ∈ S and hence S = ˆ R \ ∆ .It follows that ̟ : ˆ R → R n is univalent. For, otherwise, there were two points, p, q ∈ ˆ R such that p = q and ̟ ( p ) = ̟ ( q ). But there would be no line segment L [ p, q ]; contradiction. Moreover, for arbitrarydistinct p, q ∈ ˆ R , L [ p, q ] ⊂ ˆ R , and hence ˆ R is convex. Thus, ˆ R = co( R ) and ˆ T = T co( R ) D (cid:3) The above proof immediately implies the following generalization due to M. Abe [1].
Theorem 1.10.
Let ̟ : R → R n be a real domain over R n and let π : T R = R + i R n → C n be a domainas in (1.5) . Then, T R is a domain of holomorphy if and only if T R is univalent and convex. Notes.
Theorem 1.1 was proved by S. Bochner [2], [3], and by K. Stein [19] (Hilfssatz 1) in n = 2.Since then there have been many papers dealing with the proof (cf. Jarnicki–Pflug [18], § n = 2), S. Hitotsumatsu [8], L. H¨ormander [10] (Theorem2.5.10), etc.)(iii) By the boundary distance function (H.J. Bremermann [6] in the case of n = 2)D(iv) An approximation theorem of Bauendi-Treves (J. Hounie [9]).The present proof may belong to (iii) and was inspired by Fritzsche–Grauert [7] p. 87 Exercise 1, whilein the textbook the notion of unramified domains is presented in the subsequent section after it; so thesupposed situation might be different to the present one. It is also noticed that the observation of (1.7)goes back to Bremermann [6] § T and the convexity are proved at once.2 . Counter-examples of Abe’s Theorem 1.10 for finite tubes Here we give examples of ‘finite tubes’ by replacing the imaginary part R n in Theorem 1.10 by abounded domain, to say, an open ball, for which the theorem no longer holds.Let 0 < R < R ≤ ∞ and set A = { x = ( x , x ) ∈ R : R < k x k := ( x + x ) / < R } ,B = { y = ( y , y ) ∈ R : k y k < R } . With complex coordinates z j = x j + iy j ( j = 1 ,
2) we define a ‘finite tube’ or a ‘tube of finite length’ byΩ = A + iB ⊂ C . We consider a holomorphic function f ( z ) = z + iz ∈ O (Ω) (it is the same with f ( z ) = z − iz ). Since | f ( z ) | = | x + ix + i ( y + iy ) | ≥ | x + ix | − | y + iy | > ,g ( z ) = 1 /f ( z ) ∈ O (Ω); in particular, g ( z ) is not holomorphic at the origin 0. Therefore we first note: Remark not equal to co( A ) + iB . This give a counter-example for Kajiwara [11], from which ˆΩ = co( A ) + iB should follow. Cf. Jarnicki-Pflug [18], § ≤ ν ≤ ∞ . For 2 ≤ ν < ∞ we put A ν = n u = ( u , u ) ∈ R : R /ν < k u k < R /ν o ,p ν : A ν ∋ u = u + iu u ν = x + ix = ( x , x ) = x ∈ A, where the complex structures of ‘ u + iu ’ and ‘ x + ix ’ are different and independent to that of ( z , z ) ∈ C . It follows that p ν is a local real analytic diffeomorphism between the annuli. We put π ν : Ω ν = A ν × B ∋ ( u, y ) → p ν ( u ) + iy ∈ Ω ֒ → C . Then π ν : Ω ν → C is a local real analytic diffeomorphism and hence an unramified domain over C . Weconsider f ν ( z ) = ( f ( z )) /ν = ( x + ix + i ( y + iy )) /ν , which is ν -valued holomorphic in z ∈ Ω. Notethat f ν ( z ) = ( x + ix ) /ν (cid:18) i y + iy x + ix (cid:19) /ν :Here the latter product factor (cid:16) i y + iy x + ix (cid:17) /ν has a 1-valued branch in Ω, because(2.2) (cid:12)(cid:12)(cid:12)(cid:12) y + iy x + ix (cid:12)(cid:12)(cid:12)(cid:12) < . Whereas the first factor ( x + ix ) /ν is defined to be 1-valued in A ν , and hence f ν ( z ) is 1-valuedholomorphic in Ω ν . It follows that the domain π ν : Ω ν → C is holomorphically separable and g ν =1 /f ν ∈ O (Ω ν ).For ν = ∞ , we put p ∞ : A ∞ = { ( u , u ) ∈ R : log R < u < log R , u ∈ R } −→ A ∈ ∈ u = ( u , u ) e u e iu = ( e u cos u , e u sin u ) . Then p ∞ : A ∞ → A is a local real analytic diffeomorphism. Set π ∞ : Ω ∞ = A ∞ × B ∋ ( u, y ) p ∞ ( u ) + iy ∈ Ω ֒ → C . Then, π ∞ : Ω ∞ → C is an infinitely-sheeted unramified domain over C .3e take f ∞ ( z ) = log f ( z ). Then we have f ∞ ( z ) = log( x + ix ) + log (cid:18) i y + iy x + ix (cid:19) , z ∈ Ω :Here, because of (2.2) the second term log (cid:16) i y + iy x + ix (cid:17) has a 1-valued branch in Ω and the first termlog( x + ix ) is 1-valued in Ω ∞ , so that f ∞ ∈ O (Ω ∞ ). Therefore, the unramified domain π ∞ : Ω ∞ → C is holomorphically separable. Since f ∞ has no zero in Ω ∞ , 1 /f ∞ ∈ O (Ω ∞ ).Thus we have: Proposition 2.3.
Let the notation be as above. For every ν with ≤ ν ≤ ∞ , π ν : Ω ν → C is a ν -sheetedholomorphically separable unramified domain over C , and the envelope of holomorphy ˆ π ν : ˆΩ ν → C of Ω ν is never univalent over C and ˆ π ν ( ˆΩ ν ) .
3. Local and finite tubes
In [14], p. 32, M. Kashiwara stated the following lemma without proof, which was given by H. Komatsu[12] (here, the roles of real and imaginary parts of coordinates are exchanges from [14]):
Lemma 3.1 (Kashiwara 1970, Komatsu 1972) . Let ϕ ( z ) be a holomorphic function in a neighborhood ofthe set { z = ( z j ) = ( x j + iy j ) ∈ C n : 0 < x < a, x = · · · = x n = 0 , | y j | < b, ≤ j ≤ n } with a, b > . Then, there is a small ε > such that ϕ ( z ) is analytically (holomorphically) continued ina local convex conic open set { ( z j ) ∈ C n : 0 < x < ε, k x ′ k < εx , | y j | < ε, ≤ j ≤ n } , where x ′ = ( x , . . . , x n ) and k x ′ k = (cid:16)P nj =2 | x j | (cid:17) / . This lemma is of importance in the fundamental part of Sato’s hyperfunction theory as used andreferred as in Kashiwara [14], Komatsu [13] and Bony–Schapira [5]. The method of Komatsu’s proof [12]is a variant of H¨ormander’s [10], Theorem 2.5.10, and so belongs to (ii) of Notes at the end of §
1. Herewe consider a local version of this kind of Bochner’s Tube Theorem 1.1.Let n ≥
2. Let V be an open subdomain of the strip { x = ( x j ) ∈ R n : 0 < x < r } with a positiveconstant r , and let B = { y = ( y j ) ∈ R n : k y k < R } be an open ball of radius R ( > V + iB ⊂ C n . We then ask:
Question . Is every f ∈ O (Ω) continued analytically in co( V ) + iB ? In other words, does co( V ) + iB ⊂ ˆΩ as a univalent subdomain hold?The examples in § C n , and V and B cannot be arbitrary and need some conditions for the validity of the above Question.For 0 < τ <
1, we set τ B = {k y k < τ R } and denote by V τ a connected component of V ∩ { ( x j ) ∈ R n : 0 < x < r, k x ′ k < (1 − τ ) R } . We then consider the subdomains of Ω defined by ω τ = V τ + iB ⊃ ω ′ τ = V τ + iτ B (0 < τ < . Lemma 3.4.
Let the notation be as above. Then, ˆ ω τ contains co( ω ′ τ ) = co( V τ ) + iτ B as a univalentsubdomain.Proof. Let π : ˆ ω τ → C n be the envelope of holomorphy of ω τ . We then have that π (ˆ ω ) ⊂ co( V τ ) + iB .Let G be the connected component of π − (co( V τ ) + iτ B ) containing ω ′ τ . By Oka’s boundary distanceTheorem 1.3, − log δ ( ζ, ∂ ˆ ω τ ) is plurisubharmonic in ζ ∈ ˆ ω τ . In particular , we see that − log δ ( ζ, ∂ ˆ ω τ )( > − log((1 − τ ) R )) is a function depending only on x with π ( ζ ) = x + iy and y ∈ τ B . Let p, q ∈ G be twodistinct points such that π ( p ) = x + iy, π ( q ) = u + iv with y, v ∈ τ B , and let L [ p, q ] be the line segment4onnecting p and q in G as in the proof of Theorem 1.1, provided that it exists. Then, the restrictedplurisubharmonic function − log δ ( ζ, ∂ ˆ ω τ ) | L [ p,q ] is a convex function on L [ p, q ]. It follows thatmin L [ p,q ] δ ( ζ, ∂ ˆ ω τ ) = min { p,q } δ ( ζ, ∂ ˆ ω τ ) . The same arguments as in the proof of Theorem 1.1 in § L [ p, q ] exists forevery pair of such two points p, q , so that G is univalent and convex. Therefore, G = co( V τ ) + iτ B .For s > ≤ k ≤ n we put A k = { ( x , x , . . . , x k , x ′′ ) ∈ R k × R n − k : 0 < x < r, ≤ x j ≤ s, ≤ j ≤ k, x ′′ = 0 } . Let V ⊂ { ( x , x ′ ) : 0 < x < r } be a real domain containing A k . For 0 < τ < V τ tobe the connected component containing A k . We then have the following corollary implying Kashiwara’sLemma 3.1, Komatsu [13] Lemma 5.8, and Bony–Schapira [5] Theorem 2.2 as in the case of k = 1: Corollary 3.5.
For any < τ < , every holomorphic function f ∈ O (Ω) (cf. (3.2) ) is analyticallycontinued in the convex domain co( V τ ) + iτ B ; in particular, for < r ′ < r and < τ < there is apositive number θ such that f is analytically continued in { ( x , x , . . . , x k , x ′′ ) ∈ R k × R n − k : 0 < x < r ′ , ≤ x j ≤ s, ≤ j ≤ k, k x ′′ k < θx } + iτ B. We keep the notation. We set(3.6) ˜Ω = [ <τ< (co( V τ ) + iτ B ) . We finally have the following.
Theorem 3.7 (Finite Tube) . Let Ω be as in (3.2) . Assume that V ⊃ A k . Then the envelope ofholomorphy π : ˆΩ → C n of Ω contains ˜Ω defined by (3.6) as a univalent subdomain.Problem . It seems unknown what is the condition of V in (3.2) for ˆΩ to be univalent. For example,if V is simply connected or contractible, is then ˆΩ univalent? Remark V is convex (Kajiwara[11]). Cf. Jarnicki–Pflug [18], Chap. 3.
4. Appendix (1) Envelope of holomorphy.
In quite a few references, the notion of the envelope of holomorphy ofdomains over C n are presented in a rather sophisticated manner. For our aim the following simple-mindedconstructive existence is sufficient.We first fix a notation. If D is a connected Hausdorff space and π : D → C n is a local homeomorphism, π : D → C n or simply D is called a (unramified Riemann) domain over C n . If π is injective, D is saidto be univalent. A domain D over C n naturally admits a structure of complex manifold such that π is alocal biholomorphism; the set of all holomorphic functions on D is denoted by O ( D ).For an element f ∈ O ( D ) and a point p ∈ D there is a small polydisk neighborhood of a = π ( p ) whichis identified with a neighborhood of p , and f is written there as a convergent power series in the localcoordinate z : f p := f ( z ) = X α c α ( z − a ) α . If for two points p, q ∈ D with p = q and π ( p ) = π ( q ) there is an element f ∈ O ( D ) such that f p = f q ,then π : D → C n is said to be holomorphically separable .We fix a point p ∈ D . We consider a curve C b in C n with the initial point a = π ( p ) and the end point b ∈ C n such that every analytic function f p at a defined by f ∈ O ( D ) can be analytically continuedalong C b , and defines an analytic function, denoted by f C b ( z ), at the end point b . Let Γ denote the setof all such curves C b . If C b , C ′ b ∈ Γ are homotope through a continuous family of curves belonging to Γ,then f C b b = f C ′ b b . We denote by { C b } the homotopy class in the above sense, and write f { C b } b := f C b b .5e fix a polydisk P∆ ⊂ C n with center at the origin. For f ∈ O ( D ) and C b ∈ Γ there is a polydiskneighborhood b + r P∆ ( r >
0) of b where f { C b } b ( z ) converges. Let r ( { C b } , f ) be the supremum of such r , and let Γ † denote all of { C b } such that inf f ∈O ( D ) r ( { C b } , f ) > { C b } , { C ′ b ′ } of Γ † we define an equivalence relation { C b } ∼ { C ′ b ′ } by b = b ′ , f { C b } b = f { C ′ b ′ } b ′ , ∀ f ∈ O ( D ) . Let [ { C b } ] stand for the equivalence class, and letˆ D = Γ † / ∼ , ˆ π : [ { C b } ] ∈ ˆ D → b ∈ C n be respectively the quotient set and the natural map. It follows from the construction that ˆ π : ˆ D → C n gives rise to a holomorphically separable (unramified) domain over C n . Since D is arc-wise connected, ˆ D is independent of the choice of p ∈ D . There is a natural holomorphic map η : D → ˆ D with π = ˆ π ◦ η .If D is holomorphically separable, then η is an inclusion map and D is a subdomain of ˆ D .We call ˆ π : ˆ D → C n the envelope of holomorphy of D . In the case of n ≥
2, even if D is univalent,the envelope of holomorphy ˆ D of D may be (infinitely) multi-sheeted over C n in general. If η : D → ˆ D is biholomorphic ( D = ˆ D ), D is called a domain of holomorphy . (2) Boundary distance. The boundary distance δ ( ζ, ∂ D ) is defined as follows. For a point ζ ∈ D there is an open ball B ( π ( ζ ); r ) ⊂ C n with center π ( ζ ) and radius r ( >
0) such that the connectedcomponent U ( ζ ; r ) of π − B ( π ( ζ ); r ) containing ζ is biholomorphically mapped onto B ( π ( ζ ); r ) by π . Wewrite δ ( ζ, ∂ D ) for the supremum of such r , which is called the boundary distance . The proof of Theorem1.3 is similar to the case of univalent domains.In place of an open ball we may use a polydisk P∆ with center at 0 in the above definition. Then theboundary distance is denoted by δ P∆ ( ζ, ∂ D ); Theorem 1.3 holds with ‘ − log P∆ ( ζ, ∂ D )’. Acknowledgment.
The author is very grateful to Professor Makoto Abe for useful and helpfuldiscussions during the preparation of the present note. He also expresses a deep gratitude to ProfessorP. Schapira for kind comments on the importance of the local Bochner tube theorem due to Kashiwarafrom the viewpoint of applications to Sato’s hyperfunction theory and differential equations.
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