There are equivalent characterizations for holomorphic
functions defined on open sets of $\mathbb C^n$; first of all, they can be represented locally as sums of convergent power series. It is obvious
that a holomorphic function of several complex variables is separately holomorphic in each variable. Just separating variables, a
lot of the well-known properties of holomorphic functions of one
complex variable, as the integral Cauchy formula, have a corresponding version in several complex variables; for separation of
variables, we need the function to be continuous. Surprisingly, a
function which is separately holomorphic, is indeed C^0 and even
C^1 and therefore holomorphic (Hartogs Theorem, 1906).
This short note deals with the problem of separate analyticity
and extends the discussion to the case of separately CR functions
defined on CR manifolds. We present our result of [5] and explain
how it is related to the former literature. In particular, we explain
its link with former results by Henkin and Tumanov of 1983 and
by Hanges and Treves of 1983

Mascolo, Raffaella

OpenstarTs

Rend. Istit. Mat. Univ. TriesteVol. XXXIX, 395–406 (2007)Variations on Hartogsand Henkin-Tumanov TheoremsRaffaella Mascolo (∗)Summary. - There are equivalent characterizations for holomorphicfunctions defined on open sets of Cn; first of all, they can be rep-resented locally as sums of convergent power series. It is obviousthat a holomorphic function of several complex variables is sepa-rately holomorphic in each variable. Just separating variables, alot of the well-known properties of holomorphic functions of onecomplex variable, as the integral Cauchy formula, have a corre-sponding version in several complex variables; for separation ofvariables, we need the function to be continuous. Surprisingly, afunction which is separately holomorphic, is indeed C0 and evenC1 and therefore holomorphic (Hartogs Theorem, 1906).This short note deals with the problem of separate analyticityand extends the discussion to the case of separately CR functionsdefined on CR manifolds. We present our result of [5] and explainhow it is related to the former literature. In particular, we explainits link with former results by Henkin and Tumanov of 1983 andby Hanges and Treves of 1983.1. IntroductionIn complex analysis in several variables, the study of the propertiesof holomorphic functions on domains of Cn shows great differencesfrom the case of one complex variable. In particular, the theme(∗) Author’s address: Raffaella Mascolo, Dipartimento di Matematica Pura edApplicata, Via Trieste, 63 - 35121 Padova; E-mail: mascolo@math.unipd.itKeywords: Hartogs Theorem, CR Functions, Separate Analyticity.396 R. MASCOLOof separate analyticity has meaning only in the setting of severalcomplex variables.Let us recall some properties of holomorphic functions in the caseof one complex variable: the real and imaginary parts of an analyticfunction are harmonic functions, which are conjugate to each other; aharmonic function is uniquely determined by its boundary values; wecan construct a harmonic function with prescribed boundary valuesand construct locally its harmonic conjugate, which is unique up toan additive constant.In actual research, several complex variables play a more crucialrole. Their theory is quite difficult to treat, compared with the the-ory of one complex variable; the real and imaginary parts of an ana-lytic function are now pluriharmonic; this fact imposes a restrictionstronger than being merely harmonic, and it is not always possible toconstruct a pluriharmonic function with prescribed boundary valueson a given part of the boundary.The most relevant phenomenon, which makes the difference be-tween the cases of one and several variables, is connected to theconcept of domain of holomorphy (this is a domain Ω in Cn whichis the natural domain of a holomorphic function, so there does notexist Ω1 ⊃ Ω, where all the holomorphic functions on Ω extend). InC, all the domains are domains of holomorphy, while it is not truethat every domain in Cn is the natural domain of a holomorphicfunction.On a domain of C, we can easily construct a holomorphic func-tion, which is singular at a point of the boundary, while this is notalways possible in Cn. In fact, there exist domains Ω̃ in Cn such thatall holomorphic functions holomorphically extend to Ω̃1 ⊃ Ω̃.The note is divided into four parts: in the first part, after thedefinition of holomorphic and separately holomorphic function on adomain of Cn, we introduce CR functions on CR submanifolds of Cn.The second part is used to present some properties of holomorphicfunctions in several complex variables, as Cauchy integral formula,to give the first results on separate analyticity with the hypothesisof continuity and boundedness on compacts.Then, the third part introduces Hartogs Theorem (1906), whichis really strong, because it does not require any hypothesis for a sep-VARIATIONS ON HARTOGS etc. 397arately holomorphic function to be jointly holomorphic; and finally,in the last part, we explain the setting of separately CR functionsand we regain and generalize a result of Henkin and Tumanov (1983),using the technique of approximation.2. Holomorphic functions and CR functionsWe want to consider functions f : Cn → C. Let(x, y) 7→ (z, z̄) = (x + iy, x − iy)be the identification of R2n with the diagonal of Cn × C̄n and use(z, z̄) as coordinates in Cn.Definition 2.1. A function f , defined on a domain Ω in Cn, isholomorphic if it is C1 and satisfies the differential system ∂z̄jf = 0,∀j = 1, ..., n.Definition 2.2. A function f , defined on a domain Ω in Cn, isseparately holomorphic if it satisfies the differential system ∂z̄jf = 0,∀j = 1, ..., n.This differential system is the well-known Cauchy-Riemann sys-tem. It is an immediate remark that a holomorphic function is sepa-rately holomorphic; what we want to show is that the hypothesis ofseparate analyticity suffices to conclude that the function is C1, soit is jointly holomorphic.We want to introduce CR manifolds and CR functions as gener-alizations of complex manifolds and holomorphic functions.The starting point is to notice that, given a smooth submanifoldM of Cn, its tangent space at a point z ∈ M is not invariant, ingeneral, under the moltiplication for i; so it makes sense to look forthe largest i-invariant subspace of TzM .Definition 2.3. For a point z ∈ M , the complex tangent space ofM at z is the vector space T Cz M = TzM ∩ iTzM .While TM is a bundle, it is not always true that the dimensionof T CM is constant.398 R. MASCOLODefinition 2.4. M is called a CR submanifold of Cn if the dimen-sion of T CM is constant.Example 2.5. In Cn, any complex submanifold is a CR submani-fold, because, for a complex submanifold M , the real tangent spaceis already i-invariant, so TzM = TCz M ; another example of CR sub-manifold is the class of real hypersurfaces in Cn. Instead, if weconsider the manifold M = {z ∈ Cn : |z| = 1, Im z1 = 0}, whichis the equator of the unit sphere in Cn, this is not a CR submani-fold of Cn, because at every point z 6= (±1, 0, . . . , 0) in M we havedimCTCz M = n − 2, while, at z = (±1, 0, . . . , 0) ∈ M , TzM is i-invariant, so dimCTCz M = dimCTzM = n − 1.We give other definitions• T 1,0(Cn) := Span{∂zj} and T0,1(Cn) := Span{∂z̄j}.• M is called totally real if T Cz M = {0}, for every z ∈ M .• M is called generic if TzM + iTzM = TzCn, for every z ∈ M .• T 1,0M := T 1,0Cn∩(C⊗TM) and T 0,1M := T 0,1Cn∩(C⊗TM).Now, for a CR submanifold M of Cn, it is possible to introduce thenotion of CR function on M . Let M be a generic CR submanifold ofCn, defined by a system ρ1 = 0, . . . , ρd = 0 of independent equations.The following two definitions are equivalentDefinition 2.6. A C1 function f : M → C is CR if L̄f = 0, forevery L̄ ∈ T 0,1M .Definition 2.7. A C1 function f : M → C is CR if ∂̄f̃ ∧ ∂̄ρ1∧ . . .∧∂̄ρd = 0 on M , where f̃ : Cn → C is any C1 extension of f .CR functions on CR manifolds are analogous to holomorphicfunctions on complex manifolds, though there are relevant differ-ences, as the fact that CR functions are not always smooth. For theanalogies, the first is that the restriction of a holomorphic function toa CR submanifold is a CR function; in particular, if M = Cn, holo-morphic functions are CR, while, for the converse, we need M and fto be Cω. In general, the class of CR functions is strictly larger thanthe class of restrictions of holomorphic functions. In other terms,CR functions not always extend as holomorphic functions.VARIATIONS ON HARTOGS etc. 399Example 2.8. M = R × R is a totally real and generic submani-fold of C2 and T 0,1M = {0}, so all the functions f(x1, x2) of classC1 on M are CR. M = R × C is a generic submanifold of C2, suchthat T 0,1M is spanned by the vector field ∂z̄2 . Thus every functionf(x1, z2) of class C1 on M , which satisfies ∂z̄2f = 0, is CR. Thesefunctions are separately holomorphic in z2 and the holomorphic ex-tension needs f to be Cω. Finally, M = C×C is a complex subman-ifold and T 0,1M is spanned by ∂z̄1 and ∂z̄2 ; thus, CR functions andholomorphic functions coincide in this case.3. Properties of holomorphic functions of severalcomplex variablesTheorem 3.1 (Cauchy integral formula on polydiscs). Let f be acontinuous function on the closure of a polydisc P = D1× . . .×Dn ⊂Cn, which is, for any j, a holomorphic function of zj, when the othervariables zk, for k 6= j, are fixed. Then, for any z ∈ P , we havef(z) = (2πi)−n∫∂0Pf(ζ)(ζ1 − z1) · · · (ζn − zn)dζ1 ∧ ... ∧ dζn,where ∂0P = ∂D1 × . . . × ∂Dn.This is the generalization of Cauchy formula on the complexplane, which is a consequence of Stokes formula. For f ∈ C1(Ω̄),where Ω is a bounded open set of C, it says that∫∂Ω fdz =∫ ∫Ω df ∧dz =∫ ∫Ω∂f∂z̄dz̄ ∧ dz. In particular, if f is C1(Ω̄) and analytic, then∫∂Ω fdz = 0.The C1-regularity in each zj is needed for Stokes formula, andthe joint C0-regularity is needed for Fubini Theorem.Corollary 3.2. If f is C0(Ω) and separately holomorphic in eachzj , when the other variables are fixed, then f is C∞(Ω). (In partic-ular f is holomorphic on Ω.)It suffices to consider Cauchy formula on a polydisc contained inΩ; the integrand is C∞ and analytic in z when (ζ, z) ∈ ∂0P × P , sowe can derive under the integral.400 R. MASCOLOTheorem 3.3 (Cauchy inequalities). Let f be holomorphic on P andcontinuous on P̄ . Then|fα(zo)| ≤α!rαsup ∂0P (zo,r)|f |.Corollary 3.4. Let {fn} be a sequence of holomorphic functionson Ω, and {fn} converges uniformly to f on compact sets of Ω; thenf ∈ hol(Ω) and {∂αfn} converges uniformly to ∂αf on compact setsof Ω.Corollary 3.5 (Stjelties-Vitali). Let {fn} be a sequence of holo-morphic functions on Ω, uniformly bounded on compact sets of Ω;then, there exists a subsequence {fnk} uniformly convergent on com-pact sets of Ω, and its limit is holomorphic.Another result on separate analyticity is then given through thehypothesis of boundedness on compact sets, using the technique ofthe Theorem of Stjelties-Vitali: for holomorphic functions, uniformboundedness is equivalent to equicontinuity.Proposition 3.6. If f is separately holomorphic and bounded oncompact sets of Ω, then f is holomorphic on Ω.4. Hartogs TheoremTheorem 4.1 (Hartogs, 1906). If f : Ω → C is separately holomor-phic, then it is holomorphic. (We do not need any hypothesis on theinitial regularity of f ; this is C1 as a consequence.)Remark 4.2. A corresponding result for real analytic functions isfalse, as the following example showsf : R2 → Rf(x, y) ={xy/(x2 + y2) (x, y) 6= (0, 0)0 (x, y) = (0, 0).f is separately Cω in x (when y is fixed) and in y (when x is fixed),f is bounded on the plane, but f is not continuous at (0,0).VARIATIONS ON HARTOGS etc. 401The proof of Hartogs Theorem needs a result on subharmonicfunctions, which is known as Hartogs lemma.Lemma 4.3 (Hartogs lemma). Let vk be a sequence of subharmonicfunctions, which are uniformly bounded on any compact subset of Ω.Let lim supk→∞vk(z) ≤ C, ∀z ∈ Ω; then, ∀ǫ > 0 and ∀K ⊂⊂ Ω,there is k0 such thatsupz∈Kvk(z) ≤ C + ǫ, ∀k ≥ k0.The proof is an application of Fatou’s lemma in the integralswhich enter in the submean property. For sequences of functionswhich admit integral representations, or estimates by integrals likesubmeans, in case they have a uniform bound, then the pointwise“lim sup” enters into the integrals and becomes “uniform”.The proof of Hartogs Theorem can be divided into two parts:the first is an application of Baire Theorem, the second uses Hartogslemma.Proof. The statement is local and can be proved by adding, one byone, the directions of separate analyticity: so we can consider thebidisc ∆̄ × ∆̄ ⊂⊂ Ω in C2 and prove these two stepsSTEP 1: Analyticity on ∆ǫ × ∆. We prove that f , which isseparately holomorphic in z1 and z2, is holomorphic on ∆ǫ × ∆.Let us define El := {z1 ∈ ∆ : supz2∈∆|f(z1, z2)| ≤ l}. El isclosed and ∪lEl = ∆. By Baire Theorem, there exists l0 such thatInt(El0) 6= ∅; so f is holomorphic on Int(El0) × ∆ and, repeatingthe same construction with different sets El on any open subset of∆, we can say that f is holomorphic on B × ∆, for an open densesubset B ⊂ ∆. Also, we can assume, without loss of generality, that0 ∈ B, so f is holomorphic on the strip ∆ǫ × ∆.STEP 2: Analyticity on ∆ × ∆. At this point, we can evenforget that f is separately holomorphic in z2, when z1 is outside ∆ǫ.So we prove that, if f is holomorphic on ∆ǫ ×∆, and separatelyholomorphic for z1 ∈ ∆ when z2 is fixed, then f is holomorphic on∆ × ∆.402 R. MASCOLOWe consider the Taylor series of f with respect to z1, at z1 = 0:f(z1, z2) =∑k∂kz1f(0, z2)k!zk1 ;it converges uniformly for z2 ∈ ∆ and normally for z1 ∈ ∆ǫ. Wedefinevk(z2) :=(|∂kz1f(0, z2)|k!) 1k.Cauchy inequalities yield, by the assumption of separate analyticityof f in z1 ∈ ∆, for fixed z2, lim supk→∞vk(z2) ≤ 1; on the otherhand, by the assumption of analyticity of f on ∆ǫ × ∆, they yieldlim supksupz2∈∆vk ≤ ǫ−1. By Hartogs lemma, the pointwise esti-mate in z2 becomes uniformlim supk sup∆vk ≤ 1.Hence, the power series in z1, with holomorphic coefficients in z2,converges normally for z1 ∈ ∆ and uniformly for z2 ∈ ∆, so the sumis a holomorphic function on ∆ × ∆.Remark 4.4. Note that Hartogs Theorem consists only in Step 2;Step 1 is a preliminary by Baire. In the second step we prove twoimportant results: the uniformity in ∆ and the propagation. Whenwe say that “we can even forget that f is separately holomorphic inz2, when z1 is outside ∆ǫ”, even if we suppose that f is continuous,it is not easy to prove the analyticity on the bidisc: the problem hasbecome a problem of propagation.VARIATIONS ON HARTOGS etc. 403We can appreciate the proof through this figureSTEP 1STEP 2Figure 1: Hartogs Theorem5. Separately CR functionsThe first remark of this section is an application of Hartogs Theorem;we can easily prove it just iterating the technique of “doubling” theradius of convergence.Remark 5.1. Let f be separately holomorphic on ∆+ × ∆ ={(z1, z2) ∈ ∆ × ∆ : Re z1 > 0} and holomorphic on ∆+ǫ × ∆ ={(z1, z2) ∈ ∆+ × ∆ : |z1| < ǫ}; then, f is holomorphic on ∆+ × ∆.When the leaves of the foliation are complex curves, the problemchanges again; it is a problem of propagation, as Step 2 was, fora non-holomorphic foliation; in full generality, the validity of thestatement is an open question. If we suppose that f is C0, we havethe following statementLet {γλ}λ∈Λ be a foliation of ∆+ × ∆ by complex curves, such thatγλ∩(∆+ǫ ×∆) 6= ∅, ∀λ ∈ Λ. Let f be a C0 function on ∆+×∆, suchthat f is holomorphic on ∆+ǫ × ∆ and f|γλ is holomorphic, ∀λ ∈ Λ;then, f is holomorphic on ∆+ × ∆.The result can be proved in several ways. (For instance, it is alsoa corollary of the subsequent Theorem 5.6.)404 R. MASCOLOz 1∆ ε ∆Χ2z+Figure 2: Foliation by complex curves.The former statement can be generalized in many directions:first, in replacing the open set ∆+ × ∆ by a CR manifold M , and∆+ǫ ×∆ by an open set Mǫ ⊂ M (that can even be shrunk to a propersubmanifold N), and also in replacing the foliation {γλ} of complexcurves by a foliation {γλ} of CR manifolds of CR dimension 1.Theorem 5.2 (Mascolo). Let M be a CR connected manifold withboundary N , foliated by a family {γλ} of CR manifolds of CR dimen-sion 1, issued from N , with T Cγλ transversal to TN at any commonpoint of γλ ∩ N . Let f be a C0 function on M , which is CR alongN , CR and C1 along each γλ. Then, f is CR all over M .Remark 5.3. Since M is CR, then N is also CR. In fact, its CRcodimension is always ≤ 1, but it is in fact ≡ 1 because we have afoliation by leaves whose complex structure is transversal to N .Remark 5.4. M is a CR manifold of Cn; by a projection Cn →TzM + iTzM , which is a diffeomorphism when restricted to M , wecan assume without loss of generality that M is generic.An idea of the proof, which is divided into two steps, is to select atotally real manifold Eo ⊂ N , invariant under the foliation {Lλ∩N},and define an approximation of f by entire functions {fα}, definedbyfα(z) =(απ)n2∫+Eof(ξ)e−α(ξ−z)2dξ1 ∧ . . . ∧ dξn.By deforming the manifold Eo, we see that the above sequence pro-vides in fact a uniform approximation of f in a neighbourhood Mǫ ofVARIATIONS ON HARTOGS etc. 405Eo in M ; hence f is CR in that neighbourhood. (The same methodof polynomial approximation was first exploited by Tumanov in [6],in proving that a given function is CR.) A repeated use of this tech-nique and a connectedness argument yield the global theorem.NM MMεMFigure 3: CR extension from N to Mǫ, and global extension.Our Theorem is a generalization of the following Theorem byHenkin and TumanovTheorem 5.5 (Henkin-Tumanov, 1983). Let {γλ} be a foliation ofM by complex curves which are transversal to N at any commonpoint of γλ ∩ N , and let f be C0 on M , CR on N and holomorphicalong each γλ; then, f is CR on M .In turn, Theorem 5.5 is related to the followingTheorem 5.6 (Hanges-Treves, 1983). Let M be a hypersurface ofCn, Ω one side of M , γ a complex curve of M , zo a point of γ, f aholomorphic function on Ω, such that |f(z)| ≤ Cdist(z, ∂Ω)−N forsuitable C and N . Then, if f extends across M at zo, it also extendsat any other point z1 ∈ γ.On one hand, Theorem 5.5 is far more general, because what ispropagated is the property of f of being CR, not necessarily holo-morphic. On the other hand, in Theorem 5.6 we do not have anyfoliation by complex curves: there is only one leaf, which has theproperty of “being a propagator”. (Cf. [1] and [2] for a more de-tailed account on propagation; note that the techniques of [1] and[2] apply also to a function f which is not tempered at ∂Ω.)Example 5.7. In C4 we consider M = {z = (z1, z2, z3, z4) ∈ C4 :y1 ≥ −(|z2|2+|z3|2+|z4|2)}, with boundary N = {z = (z1, z2, z3, z4) ∈406 R. MASCOLOC4 : y1 = −(|z2|2 + |z3|2 + |z4|2)}. Let (γa,b,c)a,b,c∈R be manifolds inC4 defined by y2 = |z1|2 + ay3 = |z1|2 + by4 = |z1|2 + c.The setting of this example is good for our Theorem; the use ofHenkin-Tumanov Theorem is not possible, because the γa,b,c arestrictly pseudoconvex, so they can not be foliated by complex curves.References[1] L. Baracco, Extension of holomorphic functions from one side of ahypersurface, Canad. Math. Bull., 48 , no. 4 (2005), 500–504.[2] L. Baracco and G. Zampieri, Propagation of CR extendibilityalong complex tangent directions, Complex Variables, 50, no. 12(2005), 967–975.[3] N. Hanges and F. Treves, Propagation of holomorphic extendibil-ity of CR functions, Math. Ann. 263 (1983), 157–177.[4] G. Henkin and A. Tumanov, Local characterization of holomor-phic automorphisms of Siegel domains, Research Institute for Com-putational Complexes, translated from Funktsional’nyi Analiz EgoPrilozheniya, 17, no. 4 (1983), 49–61.[5] R. Mascolo, A remark on a Theorem by Henkin and Tumanov onseparately CR functions, Rend. Sem. Mat. Univ. Padova 116 (2006),315–319.[6] A. Tumanov, Thin discs and a Morera theorem for CR functions,Math. Zeit. 226 (1997), 327–334.Received May 11, 2007.

Variations on Hartogs and Henkin-Tumanov Theorems

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Variations on Hartogs and Henkin-Tumanov Theorems

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