D'Agnolo, Andrea

D'Ancona, Piero

Zampieri, Giuseppe

Andrea D'Agnolo

Piero D'Ancona

Giuseppe Zampieri

Archivio istituzionale della ricerca - Università di Padova

Original Citation:Extension of CR functions to ``wedge type'' domainsPublisher:Published version:DOI:Terms of use:(Article begins on next page)This article is made available under terms and conditions applicable to Open Access Guidelines, asdescribed at http://www.unipd.it/download/file/fid/55401 (Italian only)Availability:This version is available at: 11577/2494127 since: 2018-10-13T16:40:42ZUniversità degli Studi di PadovaPadua Research Archive - Institutional RepositoryEXTENSION OF CR FUNCTIONSTO “WEDGE TYPE” DOMAINSAndrea D’Agnolo Piero D’Ancona Giuseppe ZampieriAbstract. Let X be a complex manifold, S a generic submanifold of XR, the realunderlying manifold to X. Let Ω be an open subset of S with ∂Ω analytic, Y acomplexification of S.We first recall the notion of Ω-tuboid of X and of Y and then give a relationbetween; we then give the corresponding result in terms of microfunctions at theboundary.We relate the regularity at the boundary for ∂b to the extendability of CR func-tions on Ω to Ω-tuboids of X.Next, if X has complex dimension 2, we give results on extension for some classesof hypersurfaces (which correspond to some ∂b whose Poisson bracket between realand imaginary part is ≥ 0.The main tools of the proof are the complex CΩ|Y by Schapira and the theoremof Ω-regularity of [S-Z] and [U-Z].1. The system ∂b. Let X be a complex manifold of complex dimension n, S areal analytic submanifold of XR of dimension m (XR being the real underlyingmanifold to X), Y a complexification of S. Due to the complex structure of X weget a commutative diagramSφ−→ X↓ ↗ eφYIn this article we will assume S to be a generic submanifold of X, i.e. S ×X TX =TS +S√−1TS. In particular a hypersurface is always generic.Remark 1.1. The genericity of S implies that φ̃ is smooth. In fact one has:φ̃′(S×Y TY ) = φ̃′(TS⊕S√−1TS) = φ̃′(TS) +S√−1 φ̃′(TS) = TS+S√−1TS =S ×X TX. Where the third equality follows from φ̃∣∣∣S= φ.Due to Remark 1.1, tφ̃′(T ∗X) = Y ×X T ∗X is a sub-bundle of T ∗Y .One defines ∂b as the system of complex vector fields on Y which annihilateY ×X T ∗X.Remark 1.2. One has(1) φ̃−1(OX) = O∂bY ,(2) char(∂b) = Y ×X T ∗X.The content of this paper has been the subject of a talk given at the meeting “Deux journéesmicrolocales” held in Paris, 12-13 june 1989.Appeared in: Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. (9) Mat. Appl. 2(1991), no. 1, 35–42.12 A. D’AGNOLO P. D’ANCONA G. ZAMPIERI(Here O∂bY is the sheaf of germs of holomorphic functions annihilated by ∂b.)In fact, according to Remark 1.1 one can take as a system of coordinates in Y(zi)i=1,...,m with zi = φ̃i, i = 1, . . . , n. Then clearly ∂b = (∂/∂zn+1, . . . , ∂/∂zm)and the claim follows. In particular, since TS is preserved by φ̃′, one has(1.1) (char(∂b)) ∩ T ∗SY ∼= T ∗SX.2. A brief review on the language of tuboids. Let S ⊂WX be C2-manifolds,Ω ⊂ X an open set with N(Ω) 6= ∅ (here N(Ω) denotes the normal cone to Ω in Sof [K-S,§1.2.3]).Definition 2.1. Let γ be an open convex cone of Ω×S TSX. A set U ⊂ X is saidto be an Ω-tuboid of X with profile γ iff(1) ρ(TX \ C(X \ U,Ω)) ⊃ γ.(Where ρ : TX → TSX.)Remark 2.2. If one chooses a local coordinate system (x, y) ∈ X, S = {(x, y) :y = 0} then U is an Ω-tuboid with profile γ iff for every γ ′ ⊂⊂ γ there existsε = εγ′ , so thatU ⊃ {(x, y) ∈ Ω×V γ′ : |y| < ε dist(x, ∂Ω) ∧ 1}.(Here we identify TSX ∼= X in local coordinates.)3. A link between tuboids in Y and in X. Let S,X,Y be as in §1, let Ω ⊂ Sbe an open set with analytic boundary.Our aim is to give a relation between Ω-tuboids in Y and in X.Let U ⊂ X be an open set, γ ⊂ TSX, U ′ = φ̃−1(U) ⊂ Y , γ′ = φ̃′−1(γ) ⊂ TSY(we still denote by φ̃′ the induced map φ̃′ : TSY → TSX).Lemma 3.1. U is an Ω-tuboid of X with profile γ iff U ′ is an Ω-tuboid of Y withprofile γ′.Proof. Since Ω ⊂ S, we have Ω = φ̃(Ω).If ρ(TY \C(Y \U ′,Ω)) ⊃ γ′ then ρ(TX\C(X\U,Ω)) = ρ(TX\C(X\φ̃(U ′),Ω)) =ρ(φ̃′(TY \ C(Y \ U ′,Ω))) = φ̃′(ρ(TY \ C(Y \ U ′,Ω))) ⊃ φ̃′(φ̃′−1(γ)) = γ.If ρ(TX \ C(X \ U,Ω)) ⊃ γ then ρ(TY \ C(Y \ U ′,Ω)) = ρ(TY \ C(Y \φ̃−1(U),Ω)) = ρ(φ̃′−1(TX\C(X\U,Ω))) = φ̃′−1(ρ(TX\C(X\U,Ω))) = φ̃′−1(γ) =γ′. Using this lemma and 1, 2 of Remark 1.2 we can then claimProposition 3.2. Let U be an Ω-tuboid of X with profile γ, U ′ = φ̃−1(U), γ′ =φ̃′−1(γ). We have f ∈ OX(U) iff f ◦ φ̃ ∈ O∂bY (U ′).4. A microlocal approach. Let S, X, Y as before, Ω ⊂ S an open set withanalytic boundary (Ω locally on one side of ∂Ω).The framework of this paragraph is the microlocal study of sheaves by Kashiwaraand Schapira (cf [K-S]).We will still denote by ∂b the coherent DY -module associated to the system ofcomplex vector fields, i.e. ∂b = φ̃∗(DX).EXTENSION OF CR FUNCTIONS . . . 3In [S] Schapira defined the complex of microfunctions at the boundaryCΩ|Y = µhom(ZΩ,OY )⊗ orS|Y [m],similarly we setCΩ|X = µhom(ZΩ,OX)⊗ orS|X [2n−m].To give a relation between CΩ|X and CΩ|Y we first need to translate in the languageof derived categories the results of section 1.Proposition 4.1. One hasφ̃−1(OX) = RHomDY (∂b,OY ).Proof. φ̃−1(OX) = φ̃−1RHomDX (DX ,OX) = RHomDY (∂b,OY ), where the sec-ond equality is the Cauchy-Kowalevsky-Kashiwara’s theorem which holds since φ̃is non-characteristic for DX . We then haveTheorem 4.2.(4.1) CΩ|X ∼= RHomDY (∂b, CΩ|Y ).Proof. One has µhom(ZΩ,OX) ∼= µhom(ZΩ, φ̃!OX) due to [K-S, Corollary 5.5.6].Here one notices that both complexes are supported by Y ×X T ∗X.On the other hand by [K-S, Proposition 1.3.1] φ̃!OX = φ̃−1OX ⊗ orY |X [2m −2n] = RHomDY (∂b,OY )⊗ orY |X [2m− 2n], and the claim follows. Next, similarly to the sheaf of Sato’s hyperfunctionsBS = H0(RΓS(OY )⊗ orS|Y [m]),one sets (e.g. cf [S-T])BS|X = H0(RΓS(OX)⊗ orS|X [2n−m]).Recall that, S being generic, Hj(RΓSOX) = 0 ∀j < 2n−m, then by applying R0π∗in Theorem 4.2 we get(4.2) HomDY (∂b,ΓΩ(BS)) ∼= ΓΩ(BS|X).Letα :π−1HomDY (∂b,ΓΩ(BS))→ H0(RHomDY (∂b, CΩ|Y ))β :π−1ΓΩ(BS|X)→ H0(CΩ|X),be the canonical maps and defineSS∂b,0Ω|Y (f) = supp(α(f)), f ∈ HomDY (∂b,ΓΩ(BS)),SSΩ|X(g) = supp(β(g)), g ∈ ΓΩ(BS|X).4 A. D’AGNOLO P. D’ANCONA G. ZAMPIERICorollary 4.3. Let u ∈ ΓΩ(BS|X) thenSSΩ|X(u) = SS∂b,0Ω|Y (u ◦ φ).Note that, after [Z], there is a tight relation between this corollary and Proposi-tion 3.2.Remark 4.4. Note that HomDY (∂b,ΓΩ(BS)) are nothing but the CR functions inΩ (i.e. hyperfunction solutions of the system ∂b).5. The case of a hypersurface. Let X, S, Y , Ω as before; from now on assumemoreover S being a hypersurface of XR.In this case ṪSX is the union of two half rays, say ±γ; set ±γ ′ = φ̃′−1(±γ).Fix a point x0 ∈ ∂Ω and call X± the two connected components of X \ S nearx0.Let U be a neighborhood of Ω at x0 and let f ∈ OX (U ∩X+). In this case, usingProposition 3.2, we then get an equivalent of (4.1), (4.2) without using the resultsof §4:Proposition 5.1. f extends to an Ω-tuboid of X with profile Ω ×S γ iff f ◦ φ̃extends, as a solution of ∂b, to an Ω-tuboid of Y with profile Ω×S γ′.To prove this statement, recall that, by using [Z] we get that f (resp f ◦ φ̃)extends to a tuboid with profile γ (resp γ ′ = φ̃′−1γ) iff γ∗ /∈ SSΩ|X(b(f)) (respγ′∗ /∈ SS∂b,0Ω|Y (b(f ◦ φ̃))).In fact the latter is equivalent to b(f) ∈ π∗Γγ∗a((CΩ|X)T∗SX) (resp. is equivalentto b(f ◦ φ̃) ∈ π∗Γγ′∗a((CΩ|Y )T∗SY )). (We recall that Hj(CΩ|X)T∗SX = 0 ∀j < 0.)This last remark, together with Proposition 5.1, gives the following:(5.1) SSΩ|X(b(f)) = SS∂b,0Ω|Y (b(f ◦ φ̃)).We will make use of the following mixed version of (5.1) and Proposition 5.1:Proposition 5.2. f extends to a tuboid of X with profile Ω×Sγ iff γ′∗∩SS∂b,0Ω|Y (b(f◦φ̃)) = ∅.6. Ω-regularity. Let S be a real analytic manifold, Y a complexification of S,Ω ⊂ S an open set with analytic boundary (Ω locally on one side of ∂Ω). Let ω bethe canonical 1-form.We shall endow T ∗Y of a real symplectic structure by Re dω and T ∗SY by Im dω.We shall denote by HR and HI the corresponding hamiltonian isomorphisms.Choose coordinates (x; ∂/∂x) ∈ TS, and the dual coordinates (x;√−1 η) ∈ T ∗SY ;assume Ω = {x : ϕ > 0}.Let P (x; ∂/∂x) ∈ (EY )λ, λ ∈ ∂Ω×S Ṫ ∗SY . Set p = Reσ(P )|T∗SY , q = Imσ(P )|T∗SY .We assume that {p, ϕ} ≡ 1 (and p(λ) = q(λ) = ϕ(λ) = 0).It follows that dp ∧ dϕ ∧ Imω 6= 0 and thus one can divide q = a + ϕb with{p, a} ≡ 0.EXTENSION OF CR FUNCTIONS . . . 5Proposition 6.1. Assume that in a neighborhood of λ:(6.1){p, ϕ} ≡ 1,{ϕ, q}|{ϕ=0} ≡ 0,da 6= 0 or da ≡ 0,{b, a} ≡ 0.Assume also(6.2) b ≥ 0 for ϕ ≥ 0.Then P is Ω-regular at λ (i.e.(6.3) Hom(P,Γπ̇−1(S\Ω)CΩ|Y )λ = 0 ).(Here we still denote by P the module M = DY /DY P .)proof. We first choose coordinates x = (x1, x′), x′ = (x2, x′′) in S, (x;√−1 η) ∈T ∗SY so thatp = η1, ϕ = x1.We observe that (6.2) implies {ϕ, a} ≡ 0. Thus:q(x;√−1 η) = a(x′;√−1 η′) + x1b(x;√−1 η).Assume da 6= 0; by the trick of the dummy variable (that do not affect the conclusionof the theorem) it is not restrictive to assume da ∧ ω 6= 0.One can then change the coordinates (x′;√−1 η′) so thata = η2, b = b(x1, x′′;√−1 η),λ = (0;√−1 η0), η0 = (0, . . . , 0, 1).LetN = {x : ϕ = 0},V = {(x;√−1 η) : η2 = 0}.We note that N ×S V is regular involutive. We also recall that b ≥ 0 when x1 ≥ 0.We claim that then(6.4) −HR(−dϕ) /∈ Cλ(char(M), ṼΩ),ṼΩ being the union of the leaves of VC issued from Ω×S V and C(·, ·) the normalcone in the sense of [K-S]. In fact let (z; ζ), z = x +√−1 y, ζ = ξ +√−1 η becoordinates on T ∗Y . If Imσ(p+√−1 q) = 0 thenξ1 = η2 + x1bR − y1bI.6 A. D’AGNOLO P. D’ANCONA G. ZAMPIERIWe havebR = b|T∗SY + O((|y1|+ |y′′|)|η|+ |ξ|),thus we have for some c:x1bR + c((|y1|+ |y′′|)|η|+ |ξ|) ≥{0, x1 ≥ 0−c|x1||η|, x1 ≤ 0.It follows for a new c:ξ1 ≥ −c[|ζ2|+ |ξ′′|+ (|y1|+ |y′′|+ Y (−x1)|x1|)|η|],and hence (6.4).Finally (6.4) implies (6.3) by [S-Z], [U-Z].As for the case a ≡ 0 it can be handled by using the results on Ω-hyperbolicityinstead of Ω− V−hyperbolicity (i.e. for V = T ∗SY ).(cf [S-Z,§3].) 7. An application. Let X ∼= C2 3 (u1, u2), S 3 (x1, x2, x3) a real hypersurfaceof X, Y a complexification of S, Ω = {x : ϕ > 0} ⊂ S an open set with analyticboundary. Let x0 ∈ ∂Ω, U a neighborhood of Ω at x0, X± the two components ofX \ S near x0.In this case ∂b is a vector field p(x; ∂/∂x) +√−1 q(x; ∂/∂x). We still denote byp =√−1 q the symbol σ(∂b)∣∣T∗SY.Let γ be the half space N(X+) and γ′∗ the half ray γ′∗ = tφ̃′(γ∗). Let U be aneighborhood of Ω at x0.Proposition 7.1. Assume that the functions p, q, ϕ satisfy (6.1), (6.2) at λ = γ ′∗x0and let f ∈ OX(X+ ∩ U). Then f extends to a tuboid of X with profile Ω×S γ.Proof. Clearly b(f◦φ̃) ∈ Hom(∂b,Γπ̇−1(S\Ω)(CΩ|Y ))λ. By Theorem 6.1, λ /∈ SS∂b,0Ω|Y (b(f◦φ̃)). Then f extends to U verifying (2.5) on account of Corollary 4.3. Example 7.2. Assume that(i) S = {(u1, u2) ∈ X : uj = χj(x) +√−1ψj(x), j = 1, 2, x ∈ S},(ii) ϕ = ψ1,(iii) dχ1 ∧ dχ2 ∧ dϕ 6= 0.(iv) ∂x2ψ2 + ∂x1ψ2∂x3ψ2.By (ii), (iii), ‖∂χj/∂xi‖j=1,2;i=2,3 is non singular; one can then set χ1 = x2, χ2 = x3,ψ1 = x1.In such a case we have:∂b = ∂x1 −√−1 [∂x2 + β(x1, x2, x3)∂x3 ],for β solving: √−1 ∂x1ψ2 + ∂x2ψ2 −√−1 β + β∂x3ψ2 = 0.Setting β = ∂x1ψ2, we get:(7.1) ∂b = ∂x1 −√−1 [∂x2 + ∂x1ψ2∂x3 ].Write ψ2 = x1a(x2, x3) + x21c(x1, x2, x3) and set b = 2c + x1∂x1c. Assume {ξ2 +aξ3, bξ3} ≡ 0 (for instance take a(x2, x3) = a and c(x1, x2, x3) = c(x1), or take anya(x2, x3) and let c(x1, x2, x3) = 0).Under such hypotheses (6.1) is satisfied. If we then assume b ≤ 0 for x1 ≥ 0 and(x0;√−1 η ∼ (x0;√−1 η0), we get Ω-regularity at (x0;√−1 η0).EXTENSION OF CR FUNCTIONS . . . 7Remark 7.3. Note that if b ≤ 0 for x1 ≤ 0, we get S\Ω-regularity at (x0;√−1 η0).Thus for instance for S = {u : u1 = x2 +√−1x1, u2 = x3 +√−1x21}, Ω ={x : x1 > 0} and γ+ = N({u : Imu2 > Imu21}) then any f+ (resp g+) defined inX+ ∩W+ (resp X+ ∩W−) for W± a neighborhood of S ∩ {±Imu1 > 0}, extendsto a domain of type {u : Imu1 > 0, (respImu1 < 0)Imu21 < Imu2 < εImu1} (forγ+∗ = −√−1 dReu2 in the duality TMX × T ∗MX → R associated to −Imω).This is of course classical by Bochner’s theorem.On the contrary for S = {u : u1 = x2 +√−1x1, u2 = x3 +√−1x31} and forW± a neighborhood of S ∩{√−1u1 > 0}, one has extension for f+ (resp g−) fromX+∩W+ (respX−∩W−) to a domain of type {u; Imu1 > 0, Imu21 < Imu2 < εImu1}(resp. {u; Imu1 < 0,−εImu1 < Imu2 < −Imu21})Remark 7.4. Let S = {u : u1 = x2 +√−1x1, u2 = x3 +√−1 a(x2, x3)x1}, with∂a/∂x2 + a∂a/∂x3 = 0 and Ω = {x : x1 > 0}.We have∂b =∂∂ x1−√−1 [ ∂∂ x2+ a(x2, x3)∂∂ x3],(which corresponds to the case b ≡ 0 in Proposition 6.1). Then one gets Ω andS \ Ω-regularity at both points in T ∗SY ∩ char∂b.8. Removable singularities. Let S ⊂ X ∼= C2 be a generic hypersurface, Ya complexification of S. Let N ⊂ S be an hypersurface, generic on X, given byN = {x;ϕ(x) = 0}. Let NC be a complexification of N . Assume that, for ∂b =p+√−1 q, one has {p, ϕ} ≡ 1. For q = a+ ϕb ({p, a} ≡ 0), set V = {x; a(x) = 0}.Assume (6.1) to hold and moreover:(6.2)’ b ≥ 0 on T ∗SX (for any ϕ).Let Σ ⊂ N be such that√−1N∗(Σ) ⊂ ρ$(V ) (here we denoted by ρ and $ themaps: T ∗NCρ← NC ×Y T ∗SY$T ∗Y ).Take u ∈ ΓS\Σ(BS|Y )x0 , x0 ∈ ∂Σ.Proposition 8.1. If ±λ /∈ SS(u|S\Σ) then u extends to S at x0 to a function ũwith ±λ /∈ SS(ũ).Sketch of the proof. We can look at u as being a section ofHomDY (∂b,ΓS\ΣBS|Y )x0 .Let ϕ = x1, let Ω± = {±x1 > 0} and denote by γ±(u) be the traces of u on N . Wehave SS(γ±(u) ⊂ ρ$−1SS∂b,0Ω (u) and so, by Proposition 6.1, ρ(λ±) /∈ SS(γ±(u)).Hence also ρ$−1(V ) ∩ SS(γ±(u)) = ∅.Since char(∂b) ∩ ρ−1ρ$−1vC ⊂ T ∗SY , then SS(γ±) ∩ ρ$−1(V ) = ∅. Since γ+ −γ− = 0 on S \ Σ, we can propagate by the classical sweeping-out theorem. References[B-P] M.S. Baouendi, L. Preiss Rothschild, Normal forms for generic manifolds and holomorphicextension of CR functions, J. Differential Geometry 25 (1987), 431–467.[B-T] M.S. Baouendi, F. Treves, A property of the functions and distribution annihilated bya locally integrable system of complex vector fields, Annals of Mathematics 113 (1981),387–421.[B-T] M.S. Baouendi, F. Treves, About the holomorphic extension of CR functions on real hy-persurfaces in complex space, Duke Math. J. 51, 1 (1984), 77–107.[K-S] M. Kashiwara, P. 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Zampieri, Second microfunctions at the boundary, Publ. RIMS, Kyoto Univ.26 (1990), 205–219.[Z] G. Zampieri, Tuboids of Cn with cone property and domains of holomorphy, Proc. JapanAcad. Ser. A 67 (1991).A. D’A. and G .Z.: Dipartimento di matematica pura ed applicata, via Belzoni 7,35131 Padova, ItalyP. D’A.: Scuola Normale Superiore, piazza dei Cavalieri 7, 56100 Pisa, Italy

Extension of CR functions to ``wedge type'' domains

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