ON HOLOMORPHIC EXTENSION OF FUNCTIONS ON SINGULAR REAL HYPERSURFACES IN C

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ON HOLOMORPHIC EXTENSION OF FUNCTIONS ON SINGULAR REAL HYPERSURFACES IN C

TEJINDER S. NEELON (Received 3 January 2000)

Abstract.The holomorphic extension of functions defined on a class of real hypersur- faces inCⁿwith singularities is investigated. Whenn=2,we prove the following: every C¹function onΣthat satisfies the tangential Cauchy-Riemann equation on boundary of {(z,w)∈C²:|z|^k< P(w)},P∈C¹,P ≥0 andP ≡0, extends holomorphically inside provided the zero setP(w)=0 has a limit point orP(w)vanishes to infinite order. Fur- thermore, ifPis real analytic then the condition is also necessary.

2000 Mathematics Subject Classification. 32D15, 32V05.

1. Introduction. The problem of holomorphicextensions of CR functions on non- singular real hypersurfaces ofCⁿis classical and well investigated. See, for example, [2]. An optimal result inC²was obtained by Trepreau [6] (seeSection 4).

In this paper, the phenomenon of holomorphicextension of functions on singular real hypersurfacesΣis investigated. In analogy with the nonsingular case, we need to have the notion of tangential Cauchy-Riemann equations onΣ. More precisely, we need the notion of a CR (Cauchy-Riemann) function. We call a function onΣa CR function if it is continuous onΣand satisfies tangential Cauchy-Riemann equations on the nonsingular part ofΣ.

The hypersurfaces considered in this paper are the ones that bound regions which can be swept with analytic disks, with singularities being precisely the points where these disks degenerate into points. Following [3], the approach then is to integrate the given CR functionfon the boundaries of these disks with the Cauchy Kernel, and obtain a holomorphicfunction ˜f defined in the interiors of the disks. The question then is, are the boundary values of ˜fgiven byf? The main result roughly states that, if the degenerate disks are parameterized by a set which is a uniqueness set for holo- morphic functions in the direction transversal to analytic disks, then the function ˜fis indeed the holomorphicextension off. Whenn=2,this condition is also necessary in the real analyticcase.

2. The main result. LetΩbe a region inCⁿcontaining 0. LetΣ= {z∈Ω:ρ(z)=0}

be a connected(2n−1)-dimensional subset ofΩ, whereρis a real-valuedC²function onΩsuch thatρ(0)=0 anddρ≡0 onΣ. Forz=(z1,...,zn)∈Cⁿ, letz=(z2,...,zn), and letπdenote the projectionπ(z)=z. For each fixedz₀∈π(Ω), put

γz₀= z1:ρ

z1,z₀

=0

. (2.1)

We assume thatΣsatisfies the following hypothesis: eachγ_z0either degenerates into

a point or it defines a closed connectedC¹curve in thez1-plane such that ρz1

z1,z0

=0 ∀z1∈γz₀. (2.2)

Put SΣ=

p∈Σ:dρ(p)=0 , DΣ=

z∈π(Ω):γzdegenerates

. (2.3)

Lemma2.1. IfΣ, DΣ, andSΣare as above, then the following hold:

DΣ=π

z∈Σ:ρz¯1(z)=0

, (2.4)

DΣ=π(Ω), (2.5)

π SΣ

⊂DΣ. (2.6)

Proof. Supposez₀∈DΣ. If{z⁰₁} =γz₀then(z₁⁰,z₀)∈Σ, and ifρ¯z1(z⁰₁,z₀)=0 then the equationρ(z1,z0)=0 defines a one-dimensional curve nearz1=z1⁰, which is a contradiction.

Conversely, assumez⁰∈Σis such thatρz¯1(z⁰)=0. Sinceρis real,ρz1(z⁰)=0 also.

Ifz0=π(z⁰)∉DΣthen, by the hypothesis onΣ,ρ(z1,z0)=0 is the defining equation of a closed connectedC¹curve inz1-space, which is a contradiction sincedρ(·,z₀)≡0.

To prove (2.5), assume thatDΣ=π(Ω). Let α(z)denote the unique solution of ρ(α,z)=0. Note thatαis at least continuous. Now since the map

z → α

z ,z

(2.7) parameterizesΣ, the real-dimension of Σis 2n−2. This is a contradiction to our hypothesis thatΣis(2n−1)-dimensional.

Finally, (2.6) follows immediately from (2.4).

The vector fields

Lj=ρz¯j

∂

∂z¯1−ρz¯1

∂

∂z¯j, 2≤j≤n, (2.8)

have continuous coefficients which vanish precisely at the singular setSΣ, and form a basis of tangential Cauchy-Riemann equations on the nonsingular part ofΣ.

Let Σ^± denote the two “sides” of Σ. By replacing ρ by−ρ, if necessary, we may assume that

Σ⁺=

z∈Ω:ρ(z) <0

(2.9) is the side that contains the interiors of the curvesγz.

Definition2.2. AC¹functionfdefined onΣis called a CR function if it satisfies, Ljf≡0 on the regular part ofΣ∀j=2,...,n. (2.10) (Here by aC¹function onΣwe mean a function continuous onΣ, andC¹onΣ−SΣ.)

Lemma2.3. Ifuis a CR function onΣ, then

˜ u

z

=

ρ(ζ,z)=0u ζ,z

dζ (2.11)

is a holomorphic function inπ(Ω).

Proof. It is clear that ˜uis continuous and

˜ u

z

=0 onDΣ. (2.12)

By Rado’s theorem [5] and by Hartogs’s theorem, it is enough to prove that ˜uis sepa- rately holomorphicinπ(Ω)−DΣ. It is enough to show that it is holomorphicin one of the variables, sayz2, with the remaining variables fixed. Denotez=(z3,...,zn) and fix z=z₀. For the sake of convenience of notations, in the rest of the proof ofLemma 2.3, we omit thez₀ and writeρ(z1,z2)forρ(z1,z2,z₀). Observe that the induced tangential Cauchy-Riemann vector field onρ(z1,z2)=0 is

L2=ρz¯2 ∂

∂z¯1−ρz¯1 ∂

∂¯z2. (2.13)

LetCbe an arbitrary closed rectifiable Jordan curve in thez2-space such that(z2,z0)∉ DΣfor allz2∈int(C), the closure of the interior. Ifz1=γ(θ,z2)denotes theC¹curve ρ(z1,z⁰₂)=0, then

ρ γ,z2

≡0. (2.14)

By Morera’s theorem, to prove the holomorphicity of ˜uit is enough to show that

Cudz˜ 2=0. (2.15)

If we viewdz1andd¯z1asdγandd¯γ, restricted toΣ, respectively, then onΣwe have

ρz1dz1+ρz¯1d¯z1+ρz2dz2+ρz¯2dz¯2=0. (2.16) Sinceρz¯1=0, we have

d¯z1∧dz1∧dz2= −ρ¯z2

ρ¯z1

d¯z2∧dz1∧dz2. (2.17) SinceL2u=0, the above equation implies that

d

u(z)dz1∧dz2

=

∂u

∂z¯1d¯z1+∂u

∂z¯2d¯z2

dz1dz2= 1 ρz¯1

L2u

d¯z2dz1dz2=0. (2.18) Hence the restriction of the formu(z)dz1∧dz2toΣis closed. By applying the Stokes’

theorem, we have

Cudz˜ 2=0. (2.19)

Definition2.4. Σis said to be extendible to the sideΣ⁺if the following holds: for every CR functionfonΣ∩Ω, there is a functionFholomorphicinΣ⁺and continuous onΣ⁺such thatF|Σ∩Ω=f.

By a uniqueness setEof a connected and simply connected setU, we mean a subset E ⊆U¯ such that for any function f continuous on ¯U and holomorphicin U that vanishes onEvanishes identically onU.

Theorem2.5. LetΣandDΣbe as described above. IfDΣis a uniqueness set ofπ(Ω), thenΣis extendible toΣ⁺.

Proof. Letfbe a CR function onΣ. Following [3], we define f (z)˜ =(2πi)⁻¹

ρ(ζ1,z)=0

f ζ1,z

ζ1−z1 dζ1. (2.20)

Since forz∉Σ, the integrand in ˜f is a CR function onρ(ζ1,z)=0 we can apply Lemma 2.3to conclude that ˜fis holomorphicin the domainΩ−Σ.

It needs to be shown that ˜f is an extension of f toΣ⁺. It is enough to show this for each fixedz=z₀. In order to show that ˜fis a holomorphicextension off to the inside ofρ(z1,z₀)=0, all we need to show is that ˜f (z1,z)=0, for all(z1,z)such thatz1∉int(γz). (See [4].)

Letπ1denote the projection ofCⁿto the first coordinate:π1(z)=z1. Let{Kj}^∞_j=1 be a sequence of exhausting sequence of compact open subsets ofπ1(Ω). For each j≥1,put

Ωj=Kj×π(Ω). (2.21)

We first show that the restriction of f toΣ∩Ωj extends holomorphically to Σ⁺_j = Ωj∩Σ⁺. LetR >0 be a large constant such that the following is satisfied:

z1> R ⇒ z1,z

∉Ωj ∀z∈π(Ω). (2.22)

Letz⁰₁,|z⁰₁|> R, be fixed. By (2.22) and byLemma 2.3, it follows thatz→f (z˜ ⁰₁,z)is holomorphicinπ(Ω), and it clearly vanishes onDΣ. By the hypothesis thatDΣis a uniqueness set ofπ(Ω), ˜f (z⁰₁,z)must vanish everywhere inπ(Ω). Hence

f˜ z1,z

≡0 ifz1> R . (2.23) Since ˜f is holomorphicinρ(z) >0, ˜f (z1,z)=0 for allz1∈Kjand outsideγz.

If ˜fjdenotes the holomorphicextension offtoΩ^(j)+ . It is clear from the construction of ˜fj’s that ˜fj=f˜jifj> j. By lettingj→ ∞, we get the extension to all ofΩ.

3. The two-dimensional case

Theorem3.1. LetΩ⊂C²be a region containing the origin. LetP∈C¹(πw(Ω))such thatP≥0andP≡0. Letk >1. Consider the following three-dimensional set:

Σ=

(z,w)∈Ω:|z|^k=P(w)

, (3.1)

then, in order forΣto be extendible to

(z,w)∈Ω:|z|^k< P(w)

, (3.2)

it is sufficient that either the zero serP(w)=0has a limit point inπw(Ω)orP(w)is flat at some point inπw(Ω). Furthermore, ifPis real analytic then the above condition is also necessary.

Proof. The vector field

L=2Pw¯ ∂

∂z¯−kz|z|^k−1 ∂

∂w¯ (3.3)

can be taken as a CR vector field onΣ. The intersection of each complex linew=w0

withΣis a circle, possibly degenerate, centered at 0 in thez-plane. The degenerate

ones are given by thosew∈DΣ= {w∈πw(Ω):P(w)=0}. Hence, ifP(w)=0 has a limit point, thenΣsatisfies the hypothesis ofTheorem 2.5.

Now assume thatDΣis discrete andPvanishes to infinite order atw0∈DΣ. For the sake of convenience we assume thatw0=0.

As in the proof ofTheorem 2.5, the function f (z,w)˜ =(2πi)⁻¹

|z|=p(w)

f (ζ,w)

ζ−z dζ (3.4)

is an holomorphicextension off to|z|< p(w), provided Jm(w)=

|z|=p(w)f (ζ,w)ζ^mdζ≡0 ∀m=0,1,2,.... (3.5) By following the reasoning in the proof ofLemma 2.3, and by using Rado’s theorem, we can conclude that theJm’s are holomorphicfunctions inπw(Ω). We show that the Jm’s vanish to infinite order at 0.

Letmbe fixed. RewritingJmas Jm(w)=i

p(w)_m+12π

0 f

p(w)e^iθ,w

e^i(m+1)θdθ, (3.6) we have

Jm(w)≤p(w)^m+1_2π

0

f

p(w)e^iθ,wdθ. (3.7) The integrand inJmis bounded whenwstays in a bounded set containing 0. SinceP vanishes to infinite order at 0, for every integerl >0 there is a constantClsuch that

|P(w)|< Cl·|w|^l. We have

J(w)< C_l^1/k·|w|^l(m+1)/k(2π) sup

|w|<R,θ

f

p(w)e^iθ,w< C_l|w|^l, (3.8) whereC_l>0 is a constant, andlis an integer,l→ ∞asl→ ∞. Hence,Jmvanishes to infinite order at 0.

SupposeP is real analytic. First assume thatP(w)=0 forw=0. SinceP is real analytic, there are constantsC >0 andδ >0 such that|P(w)| ≥C|w|^δ.

Let mbe a large fixed positive integer, and let u be a function onΣ defined as follows:

u(z,w)=w^m z^k

Σ if 0=(z,w)∈Σ,ifu(0,0)=0. (3.9) Since

|u| =|w|^m

|z|^k

_Σ=|w|^m

P(w)≤ |w|^m−δ, (3.10) it follows that it is continuous onΣ. By choosingmlarge, we can makeuaC^rfunction for a givenr. Clearlyuis a CR function that does not extend holomorphically to the side|z|^k< P(w). This example can be easily modified to the case when P(w)has more than one isolated zero.

4. Examples. A theorem of Trepreau [6] says that if a nonsingular real hypersurface Σ⊂C², 0∈Σ, does not contain a holomorphic curve passing through 0, thenΣis

extendible, near 0, to at least one side. The analogue of this theorem does not hold for singular real hypersurfaces as shown by the following example.

Example4.1. Consider the hypersurface Σ=

(z,w)∈C²; |z|²= |w|²+|w|⁴

(4.1) with an isolated singularity at the origin. It follows fromTheorem 3.1, thatΣis not extendible to either side. SupposeVis a holomorphic curve contained inΣ. Ifp=(0,0) is a nonsingular point of V, thenΣ is of infinite type at p (see [1]). But a simple calculation shows thatΣis of type 2 at all pointsp=(0,0). Hence,Σdoes not contain a holomorphiccurve.

Example4.2. Theorem 3.1shows that the hypersurface Σ=

(z,w)∈C²:|z|²=w²+w¯²

(4.2) is extendible to the side|z|²< w²+w¯². As before it is easy to see that the restriction of z⁴w⁻¹ to Σ is a CR function which clearly does not extend holomorphically to

|z|²> w²+w¯².

Example4.3. The hypersurfaceΣ= {(z,w)∈C²; |z|²=e^−1/|w|2}∪{(0,0)}has an isolated singularity at the origin and, byTheorem 3.1, is extendible to|z|²< e^−1/|w|2.

References

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Tejinder S. Neelon: Department of Mathematics, California State University San Marcos, San Marcos, CA92096, USA

E-mail address:[email protected]