A remark on a Theorem by Henkin and Tumanov on Separately CR Functions.
RAFFAELLA MASCOLO(*)
ABSTRACT- By using the approximation of CR functions by polynomials, we prove that functionswhich are separately CR are jointly CR. We regain in thisway a result by Henkin and Tumanov [2] by a simple and expressive proof.
I wish to thank Dr. Luca Baracco for fruitful discussions.
Our discussion starts from the following statement
Let U and V be open subsets of Cn;with UV and V connected, let (gl)l2Lbe complex curves foliating V;such thatgl\U6 ; 8l2L. If f is a C0 function defined on V; such that fjU is holomorphic and fjgl is holo- morphic8l2L, then f is holomorphic on V.
A key point in the above assumption is thatf is supposed to beC0from the beginning; the sole assumption of separate analyticity along the curves gl would not suffice for the conclusion, in general. According to Hartogs theorem, it suffices in presence of a holomorphic foliation. Otherwise, in full generality, the validity of the statement is an open question. There are many possible ways to prove the above result. For instance, if we suppose in addition thatf isC1 we have an immediate proof. Let the foliation be described by a mapping
F:CL!Cn; (t;l)7!F(t;l);
smooth in both its arguments and holomorphic int, so that the leaves are defined bygl: fF(t;l): 8t2Cg.
Since the statement is local with respect tol, we can assume that in a neighborhood of a fixed value ofl, the setUcontainsthe image underFof a neighborhood oft0. We then consider the formF(df^dz1^:::^dzn);
(*) Indirizzo dell'A.: Dipartimento di Matematica Pura ed Applicata Uni- versitaÁ di Padova Via Trieste, 63, 35121 Padova.
we have
the coefficientsof the above form are holomorphic with respect tot forlconst,
the coefficientsare0forjtj<e.
Hence the above form is0. The above argument can be put in a ``weak sense'' and in this way we can handle the case in whichf isfor instanceC0. We want to generalize the former statement in many directions: first, in replacing the pairUVbyNMwhereNandMare CR manifoldsand also in replacing the foliationfglgby a foliationfLlg by CR manifoldsof CR dimension 1. Our eventual goal is to prove that iffisC0inM, CR onN, CR andC1 along each leafLl, then it isin fact CR onM. An idea of the proof could be in selecting a totally real manifoldEoN, invariant under the foliation fLl\Ng and in defining an approximation of f by entire functionsffagdefined by
fa(z) a p
n2 Z
Eo
f(j)e a(j z)2dj1^:::^djn: 0:1
By deforming the manifoldEowe see that the above sequence provides in fact an uniform approximation off in a neighborhood ofEoinM: by us ing differentEo's, we have in particular thatf isCR overM. We will carry out in detail the above argument and get the proof of
THEOREM 1. Let M be a CR manifold of Cn, with boundary N (codimMN=1) and let (Ll)l2L be a foliation of M with the following properties: every Llis a CR manifold of CR-dimension1, the intersection of Ll with N is not empty and TCLl
N\LlTN
N\LlTM
N\Ll. If f 2C0(M)\CR(N)\CR(Ll)and f is C1along each leaf Ll, then f is, in a neighbourhood of each point of N, a limit of polynomials. In particular, it is CR in a neighbourhood of N in M.
Note thatMisnot required to be compact.
PROOF. We fix a point zo2N and prove the conclusion in a neigh- bourhood of zo. By a projection Cn!TzoMiTzoM, which isa diffeo- morphism when restricted toM, we can assume without loss of generality thatMisgeneric. We chooseEo, a totally real maximal submanifold ofN invariant under the foliationfLl\Ngand define a sequenceffagby means of the convolution (0.1) with the heat kernel. It is classical thatfa!!f as a! 1uniformly over (compact subsets of)Eo. We want to prove that in fact fa!!f in a neighborhood of Eo onM. Let ussee how to prove it.
Without loss of generality, we can assume, apart from a change of co- ordinates, thatEoisa small perturbation ofRn. We insertEointo a foliation fEgby totally real maximal submanifolds onNinvariant under the mani- foldsfLl\Ng. We denote byfSgthe family of the unionsof theLl's issued from eachE; thisprovidesa foliation ofM. Givenz2MnEo, thisbelongsto a uniqueSSz. We take a deformationE~oofEowhich containszand of the typeE~oE1[E2[E3whereE2N,E3 SzandE1 isthe piece ofEo
outside a neighborhood ofzo. We require that the deformation is small so that the following condition isfulfilled for somek<1:
j=m(j z0)jGkj<e(j z0)j 8j2E~o:
(E~oneeds possibly to be shrunk here.) We denote bySa piecewise smooth manifold contained inN[Szwith boundaryE~o Eo. We define:
fea(z) a p
n2 Z
E~o
f(j)e a(j z)2dj1^:::^djn:
The previousestimate guaranteesthatfea(z) convergestof(z)8z2E~o. We show thatfea(z)f(z)8z2M:EoandE~odelimit a submanifold ofM, that we have denoted byS. By Stokestheorem:
fa(z) fea(z) a p
n2 Z
Eo
f(j)e a(j z)2dj1^:::^djn
a p
n2 Z
E~o
f(j)e a(j z)2dj1^:::^djn
a p
n2Z
S
dSf je a j z2dj1^:::^djn :
(Note here that if f isonly C0 and not C1, as``formally'' required by StokesTheorem, nonethelessthe form dS() hasC0 coefficientsand the above conclusion holds.) To prove it, let us use the notation dS=@S+@S. Consider the term on the right: sincee a(j z)2isholomorphic (inCn) andf isCR on S, we have that the product isa CR function on S, s o
@S(f(j)e a(j z)2dj1^:::^djn)0. On the other way, @S(f(j)e a(j z)2 dj1^:::^djn)0, because the expression is full in the holomorphic differentials. Thenfacoincideswithfea, and it followsthatfaconvergestof on any perturbationE~oofEo, s ofaconvergestof in a neighbourhood ofEo
onM. p
The same method of polynomial approximation, as the one used in the above theorem, wasfirst exploited by Tumanov in [3], in proving that a given function isCR.
A repeated use of Theorem 1 and a connectedness argument yield the following global Theorem. This is substantially due to Henkin-Tumanov [2]
but it allows general foliations by manifolds of CR dimension 1 instead of complex curves. The proof is also far different.
THEOREM 2. Let M be a CR connected manifold with boundary N, foliated by a familyfLlgof CR manifolds of CR dimension1issued from N, with TCLltransversal to TCN at any common point of Ll\N. Let f be a C0function on M, which is CR along N, CR and C1along each Ll. Then f is CR all over M.
PROOF. According to Theorem 1,fisCR in a neighbourhoodUofNin M. Letxbe a point ofM; we consider a family of domainsVyM,y2[0 1], withC1and CR boundaryMy=bVy, such that:
VoU
TCbVyistransversal toTCLlat any point ofbVy\Ll
bVynUM Vy Vmifm>y S
y<mVyVm
VmT
y>mVy
V13x.
We claim thatf isCR onV1, which concludesthe proof. In fact, ifyois the maximal index for whichf isCR onVyo, we can apply Theorem 1, forN replaced bybVyo; we note here that sincefisCR onVyo, then itsboundary value onbVyo isalso CR.
We conclude thatf isCR in a neighbourhood ofbVyo. SincebVyonUis compact, we conclude, by a finite covering argument, thatf isCR in some
Vmform>yo. A contradiction. p
REMARK1. Aswe have already noticed in the course of the proof of Theorem 1, by slicingSby meansof the leavesLland by applying Fubini's Theorem, we need not to assume thatf isC1and can just require that the restrictions fjLl are C1. In particular, when the Ll'sare replaced by complex curves, this comes as a consequence of the hypothesis that f is holomorphic along these curves.
Theorem 2 appliesin particular if we replaceMby an open subsetVof Cnand the manifoldsLlby complex curvesgl, and yieldsthe proof of the following
COROLLARY 1. Let V be an open domain of Cn, N a part of its boundary,fglga foliation ofV transversal toNand letf be continuous onV, CR onN and holomorphic on eachgl. Thenf is holomorphic all overV.
REFERENCES
[1] N. HANGES - F. TREVES, Propagation of holomorphic extendibility of CR functions,Math. Ann.,263(1983), pp. 157-177.
[2] G. HENKIN- A. TUMANOV,Local characterization of holomorphic automor- phisms of Siegel domains, Research Institute for Computational Complexes, translated from Funktsional'nyi Analiz Ego Prilozheniya, 17(4) (1983), pp.
49-61.
[3] A. TUMANOV,Thin discs and a Morera theorem for CR functions, Mathema- tische Zeitschrift,226(1997), pp. 327-334.
Manoscritto pervenuto in redazione il 20 settembre 2005 e in forma riveduta il 29 settembre 2006.
