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RAD´O THEOREM AND ITS GENERALIZATION

arXiv:math/9210201v1 [math.CV] 21 Nov 1992RAD´O THEOREM AND ITS GENERALIZATIONFOR CR-MAPPINGSE.M.ChirkaAbstract. A generalization of Rad´o’s theorem for CR-functions on locally Lipschitzhypersurfaces is obtained.

It is proved also that closed preimages of pluripolar setsby CR-mappings are removable for bounded CR-functions.1. IntroductionA well-known theorem of Rad´o [9] states that a continuous function f definedon a domain in C and holomorphic on the complement of its zero set f −1(0) isholomorphic everywhere.

The result is correct for holomorphic functions in Cn aswell as in the plane. It is well-known that f −1(0) can be replaced by f −1(E), E aclosed subset of zero capacity in C (see [14]).Recently J.-P.Rosay and E.L.Stout [10] have shown that an analogue of theclassical Rad´o’s theorem take place for CR-functions on a C2-hypersurface in Cnwith nonvanishing Levi form.

Then H.Alexander [1] has proved the removability inthe same situation of closed sets of the type f −1(E), E a closed polar set in C. Weimprove here these results in the following theorem which can be considered as anextension of Rad´o’s theorem to bounded CR-mappings of hypersurfaces.Theorem 1. Let Γ be a locally Lipschitz hypersurface in Cn with one-sided exten-sion property at each point, Σ is a closed subset of Γ andf : Γ \ Σ −→Cm \ Eis a CR-mapping of class L∞such that the cluster set of f on Σ along of Lebesquepoints of f is contained in a closed complete pluripolar set E.Then there is aCR-mapping ˜f : Γ −→Cm of class L∞(Γ) such that ˜f |Γ\Σ= f.We say that Γ has one-sided extension property at its point a if for an arbitraryneighbourhood U ∋a there is a (smaller) neighbourhood V ∋a and a connectedcomponent W of V \ Γ such that a ∈¯W and every bounded CR-function on Γ ∩Uextends holomorphically into W. As it was shown by Tr´epreau [15] this propertyat each point has an arbitrary locally Lipschitz hypersurfases in C2 which containsno analytic discs.

The same is true for hypersurfases of class C2 in Cn containingno complex hypersurfaces (see [15]).1991 Mathematics Subject Classification. Primary 32D15, 32D20; Secondary 32B15, 32C30.Key words and phrases.

CR-functions, removable singularities, analytic continuation.The final version of this paper will be submitted for publication in ”Mat.Sbornik”Typeset by AMS-TEX1

2E.M.CHIRKAWe show indead that the trivial extension of f by a constant on Σ is a CR-mapping on the whole Γ.Theorem 1 is true also without the supposition of the one-sided extension prop-erty, but the general case is more compicated, and this is related with analytic discsbelonging to Γ. We shall prove the general theorem in the next paper.Theorem 1 is equivalent to the following new result on the removability of sin-gularities for bounded CR-functions.Theorem 2.

Let Γ, Σ, E and f be as in Theorem 1. Then each CR-function ofclass L∞on Γ \ Σ extends to a CR-function of class L∞on Γ.Theorem 1 follows obviously from Theorem 2.

In opposite direction, given aCR-function g of class L∞on Γ \ Σ corresponds the mapping(f, g) : Γ \ Σ −→Cm+1 \ E × Cwhich cluster set on Σ is contained in the closed complete pluripolar set E × C.By Theorem 1, the map (f, g) extends to a CR-map of whole Γ, and thus its lastcomponent g extends to there as a CR-function.If Γ is the boundary of a bounded domain or if Γ admits one-sided holomorphicextension of CR-functions (say, if Γ ∈C2 contains no complex hypersurface, as in[15]) then it follows from Theorem 2 and a uniqueness theorem that the Hausdorff(2n −1)-measure of Σ vanishes, and Γ \ Σ is locally connected.We prefer to work here with the class L∞instead of C, since L∞is stable byconsidering extensions whereas the CR-extension of a bounded continuous CR-function from Γ \ Σ onto Γ is not continuous in general.The proof of Rad´o theorem for CR-functions in [10] is based on results [8] onthe holomorphic continuation of CR-functions from a part of the boundary of adomain in Cn, n ≥2 (see also [7]). In the proof of Theorem 1 we use insteadof this a geometric extension of the graph in spirit of R.Harvey and H.B.Lawson[6].

Our starting point was a generalization of the Harvey - Lawson theorem [6] onboundaries of holomorphic chaines for MC-cicles in the complement to a polynomi-ally convex compact set [4, 1985]. It can be considered as a geometric version (forCR-functions of class C1) of theorems on holomorphic continuation in [8],[10],[7].Let us specify the terminology.We say that a hypersurfase M in a smooth k-dimensional manifold M is locallyLipschitz if for every point a ∈M there is a coordinate chart (U, x), x = (x′, xk)on M such that a ∈U and M ∩U is represented as the graph xk = h(x′) ofa function h over the domain in Rk−1 which sutisfies there Lipschitz condition| h(b) −h(c) |≤C | b −c | with a constant C. Note that the Hausdorffk −1-measure (with respect to some fixed smooth metric on M) restricted to such M islocally finite, and M has tangent planes in almost every point with respect to thismeasure (Rademacher’s theorem, see e.g.

[5,3.1.6.]). Thus the integral on M for adifferential (k −1)-form ϕ with Lipschitz coefficients and with compact suppϕ ∩Mis well-defined.A point a in a locally Lipschitz M ⊂M is called a Lebesque point for a givenvector-function f of class L1loc(M) with values in RN if there is a constant ˜f(a) ∈RNsuch thatr−k+1ZM∩|x′|

RAD ´O THEOREM AND ITS GENERALIZATION3as r →0 in the chart (U, x) with x(a) = 0 described above. It is wellknown (see[5,2.9.8.]) that almost every point a ∈M is such a point and f(a) = ˜f(a) almosteverywhere.

Thus we shall assume in the further that f is defined (as ˜f(a)) on theset of its Lebesque points a only.For a locally Lipschitz hypersurfase M in a complex n-dimensional manifold Mthe notion of CR-functions of class L1loc(M) is well-defined: a function f of thisclass is a CR-function on M if RM f ¯∂ϕ = 0 for every smooth form ϕ of bidegree(n, n −2) in M with compact suppϕ ∩M.A set E ⊂Cm is called complete pluripolar, if there is a plurisubharmonicfunction ϕ in Cm such that E = {ζ : ϕ(ζ) = −∞}.2. One-sided holomorphic extensionThe problem is local, so we can assume that Γ ∋0 is represented as the graphv = h(z′, u) of some Lipschitz function in a domain in the space of variables(z1, ..., zn−1, Rezn) = (z′, u) (it is convenient to use the notation zn = u + iv).Fix a connected component Γ0 of Γ \ Σ, set f1 = f on Γ0, f1 = 0 on Γ \ Γ0and denote by Γ1 ⊃Γ0 the set of points a ∈Γ such that f1 is a CR-function in aneighbourhood of a on Γ.

We have to show that Γ1 = Γ.By the one-sided extension property, for each point a ∈Γ1 there is a neighbour-hood Va ∋a and a connected component Wa of Va \ Γ such that a ∈Wa and eachCR-function on Γ1 extends holomorphically into Wa. Shrinking Va we can assumethat the intersection of Va with each line (z′, u) = const is an interval (i.e.

Va isconvex in v-direction) intersecting Γ1. Then the union of all Wa, a ∈Γ1 is an openset of the form W + ∪W −where W + is plased over Γ (i.e.

v > h(z′, u) on W +)and W −is contained in {v < h(z′, u)}. It follows that the setW = W + ∪W −∪(W + ∩W −∩Γ1)is open, convex in v-direction, and ¯W ⊃Γ1.By a uniqueness theorem and aremovable singularities theorem the holomorphic extensions of f1 into Wa, a ∈Γ1,constitute holomorphic functions in W + and W −, and these functions extend to aholomorphic (vector-) function in W which we denote by the same symbol f1.By the construction, there is a Lipschitz function ǫ(z′, u) such that ǫ = 0 outsideof (z′, u)(Γ1), the hypersurface Γ′ : v = (h + ǫ)(z′, u) is contained in W ∪Σ1,and Γ′ \ Σ1 is a smooth (C∞or even Cω if you want).

The set (f1)−1(E) ∩W ispluripolar, so we can assume that its intersection with Γ′ \ Σ1 has the Hausdorffdimension 2n −3, in particular, it has the locally connected complement in Γ′ \ Σ.Set Σ′1 = Σ1 ∪(Γ′1 ∩(f1)−1(E) and Γ′1 = Γ′ \ Σ′1. If ǫ(z′, u) is taken sufficiantlysmall and rapidly tends to zero as (z′, u) approaches to (z′, u)(Σ1), thenf1 | Γ′1 −→Cm \ Eand the cluster set of f1 on Σ′1 is contained in E. Thus, substituting Γ onto Γ′, Σ1onto Σ′1 and Γ1 onto Γ′1 and then restoring old notations, we can assume that Γ1is smooth and the mapping f1 is holomorphic in a neighbourhood of Γ1.

4E.M.CHIRKA3. Reducing to n = 2We assume as above that Γ ∋0 is represented as the graph of some Lipschitzfunction over a domain in the space of variables z1, ..., zn−1, Rezn.

Then the vector(0, ..., 0, i) does not belong to CaΓ, the tangent cone to Γ at the point a, for alla ∈Γ. Shrinking Γ a little we can assume that the same is true for some C-linearlyindependent system of vectors ξ1, ..., ξn, ξj ̸∈CaΓ for a ∈Γ, j = 1, ..., n. Making asuitable C-linear changing of coordinates we obtain the situation when iej ̸∈CaΓfor all standart coordinate orts ej in Cn.

It follows then that for each j there is aneighbourhood Uj ∋0 such that Γ ∩Uj is represented as the graph of a Lipschitzfunction over a domain in the space of variables zk, k ̸= j, Rezj. Set U = ∩n1 Uj.We have to show thatRΓ f0 ¯∂ϕ = 0 for an arbitrary smooth (n, n −2)-form ϕwith suppϕ ⊂U.

This form is represented as Pj

[4, A4.4.]) thatZΓf0 ¯∂ϕ =Xj

As Γjk ⊂{zl = cl, l ̸= j, k} ≃C2, we obtain that it is enough to prove Theorem 1 for thecase n = 2.4. Analytic extension of the graphTo show that Σ1 is empty we assume that 0 is the boundary point of Γ1 in Γand come at last to a contradiction.The base domain G ⊂C×R can be taken bounded and convex, and the functionh(z, u) defined and with Lipschitz condition in a neighbourhood of ¯G.

Then thegraph S : v = h(z, u) over bG is a two-dimensional sphere in C2. As 0 is limitingpoint for Γ1, we can assume, that S is not contained in Σ1.

By Shcherbina’s theorem[12, 13] the polynomially convex hull ˜S of S is the graph of a continuous function˜h(z, u) over ¯G foliated in a one-parametric family of analytic discs with boundarieson S.The graph M of the map f1 over Γ1 is a smooth maximally convex 3-dimensionalmanifold in C2×Cm which boundary ¯M \M is contained in ( ˜S×Cm)∪(C2×E). Asf1 is uniformly bounded, there are closed balls B2, Bm with centers in origins suchthat ¯M \ M is contained in ( ˜S × Bm) ∪(B2 × E).

This compact set is polynomiallyconvex due to the following.

RAD ´O THEOREM AND ITS GENERALIZATION5Lemma 1. Let X1 ⊂X2 be polynomially convex compact sets and Y is a completepluripolar set in CN.

Then the set X = X1 ∪(Y ∩X2) is polynomially convex.Proof. The set Y is represented as {ζ : ϕ(ζ) = −∞} for some function ϕ plurisub-harmonic in CN.

If a ̸∈X1 ∪Y then there is a polynomial p such that p(a) = 1 and| p |< 1 on X1. Let C = sup{ϕ(ζ) : ζ ∈X1} and a positive integer s is taken sobig that | p(ζ) |s eC < eϕ(a) for all ζ ∈X1 (it is possible because a ∋Y ).

Then thefunction ψ =| p |s eϕ is plurisubharmonic in CN, ψ(ζ) < ψ(a) for ζ ∈X, and thesame is true for ζ ∈Y because ψ | Y = 0. It follows from the maximum principlefor plurisubharmonic functions on polynomially convex hulls (see, e.g.

[3]) that ais not contained in the hull of X. The rest follows from the inclusion X ⊂X2.Thus, the polynomially convex hull of the set (S × Bm) ∪(B2 × E) for B2 ⊃Sis the compact setK = ( ˜S × Bm) ∪(B2 × E),and the graph M of f1 is attached to this K. By a generalization of Harvey - Lawsontheorem in [4, Theorem 19.6.2] there is a two-dimensional (complex) analytic subsetA in C2+m \ (K ∪M) such that A ∪K ∪M is compact and M \ K ⊂¯A.5.

The projection of the extensionWe show that A is the graph of a holomorphic mapping over an open set in C2with the boundary in Γ ∪˜S. (The main difficulty here is that the projection of ¯A isas well as Γ∪˜S contained in the ball B2, the ”shadow” of B2 × E.) It is convenientto use in this Section coordinates (z′, z”) for C2 × Cm.We essentially use the pluripolarity of B2 × E and the following result due toE.Bishop [2, 11] on the removability of pluripolar singularities for analytic sets (see[4]).Lemma 2.

Let Y be a closed complete pluripolar subset of a bounded domainU = U′ × U ” ⊂Cn+m, and A is a pure p-dimensional analytic subset in U \ Ywithout limit points on U′ × bU ”. Suppose that U′ contains a nonempty subdomainV′ such that ¯A ∩(V′ × U ”) is an analytic set.

Then ¯A ∩U is analytic in U.First of all, we apply this lemma to the unbounded component U′ of C2 \( ˜S ∪Γ).Let U ” be an open ball in Cm containing Bm and Y = U∩B2×E. Then A∩(U\Y ) isan analytic set sutisfying the conditions of Lemma 2.

By the maximum principle,A is projected into B2 because its boundary is contained in K ∪M. Thus, forV′ = U′ \ B2, the set ¯A ∩(V′ × U ”) is empty (hense analytic).

It follows fromLemma 2 that ¯A∩U is analytic in U. As ¯A∩(V′ ×U ”) is empty and the projectionof ¯A ∩U into U′ is proper, the set ¯A ∩U is also empty.

Thus, we have proved thatthe projection of A into C2 is contained in the closure of the union of all boundedcomponent of C2 \ (Γ ∪˜S).Take now an arbitrary point a′ ∈Γ1 \ ˜S and show that the set A ∩{z′ = a′} isempty. This set is closed analytic in a′ × (Cm \ (E ∪a”)) where a” = f1(a′).

AsE ∪a” is complete pluripolar, its intersection with Bm is polynomially convex. As¯A∩{z′ = a′} is compact, it follows from a maximum principle on analytic sets (see,e.g., [4, 6.3.]) that the dimension of A ∩{z′ = a′} is zero, i.e.

this set is discrete.Thus, given b = (a′, b”) ∈A there is a neighbourhood U = U′ × U ” such thatthe projection of A ∩U into U′ is an analytic covering (see [4]). But dimA = 2,

6E.M.CHIRKAand there is no point in A ∩U over unbounded componenet of C2 \ (Γ ∪˜S) whichhas nonempty intersection with U′. This contradiction shoes that there is no suchpoints b, i.e.

A ∩U is empty.Let now U′ be a bounded component of C2 \ (Γ ∪˜S) such that bU′ ∩(Γ1 \ ˜S isnot empty, and a′ is a point in this nonempty set. Then (see Sect.1.) there is aneighbourhood V′ ∋a′ such that f1 is holomorphic in V′.

We have in (V′ ×Cm)\Mtwo analytic sets, A∩(V′ ×Cm) and the graph of f1 over V′ ∩U′, of pure dimension2 with the same smooth boundary M ∩(V′ × Cm). By a boundary uniquenesstheorem for analytic sets (see [4, 19.2.

]), these sets coinside. Thus, the analyticcovering A ∩(U′ × Cm) −→U′ is onesheeted over V′ ∩U′, which follows that itis one-sheeted over whole U′.

It means that A over U′ is the graph of a boundedholomorphic map, and this map is a continuation of f1 into U′. In particular, weobtain that f1 as the boundary value of this map is CR on bU′ ∩(Γ \ ˜S).In terms of components of Γ \ ˜S, it means that there are only two possibilities:either this component is contained in Γ1 or it is contained in Σ1.6.

Removability of ΣReturn to notations (z, w) for coordinates in C2 and denote by p the projection(z, w) 7→(z, u) into C2 × R.Let δ ⊂˜S be an analytic disc with the boundary in S such that p(δ) ∩p(Γ1) isnot empty. Show that p(δ) ⊂p(Γ1).Suppose it is not.

Then there is a point a ∈Σ1 such that p(a) ∈p(δ), and aconvex domain G1 ⊂G containing p(a) such that bG1 ∩p(δ) ∩p(Γ1) is not empty,say, it contains p(b) for some b ∈Γ1. By Sect.1, f1 is holomorphic in a one-sidedneighbourhood V of b convex in v-direction, as in Sect.1.

Let h1 be a Lipschitzfunction on bG1 such that its graph S1 : v = h1(z, u) is contained in Γ ∪V butdoes not contain b. We can assume v < h(z, u) on S1 ∩V (changing w onto −wand shrinking V if it is nessessary ).

Then for ht = th + (1 −t)h1, 0 < t < 1, wehave ht ≤h and ht < h over p(V ). By Shcherbina’s theorem [12, 13] polynomiallyconvex hull eSt of St : v = ht(z, u) is the graph of a continuous function ˜ht overG1 foliated in a one-parametric family of analytic discs with boundaries in St. Asht ≤h, we have ˜ht ≤˜h, and ˜ht →˜h as t →0 by continuity.Thus, for t > 0 small enough there is a disc δt ⊂eSt such that p(a) ∈δt andp(bδ) ∩p(V ) is not empty.

As v ≤h(z, u) on δt and v < h(z, u) on bδ ∩V , we havev < ˜h(z, u) on the whole δt, i.e. the disc δt is placed strongly under the hypersurfase˜S.

(This is true because the discs δt −(0, iǫ) for ǫ > 0 do not intersect ˜S, and wecan apply the argument principle. )As we proved above (with eSt instead of ˜S), each component of p(δt \ Γ) is eithercontained in p(Γ1) or it is contained in p(Σ1).

But p(δt ∩Γ) and p( ˜S ∩Γ) have nocommon point because δt ∩˜S is empty, and the projection p | Γ is one-to-one. Asp(δt) ∩p(Γ1) is not empty by the construction, the set p(δt) ∩p(Σ1) must to beempty, in particular, p(a) ̸∈p(Σ1).

The contradiction (with choosing of a) showsthat there is no such point a, i.e. p(δ) ⊂p(Γ1.If Σ1 is not empty, it follows from the above that there is a disc δ0 ⊂˜S such thatp(δ0) ⊂p(Σ1) ∩∂(p(Γ1)).

As δ0 is not contained in Γ, there is a point c ∈Σ1 \ ˜Ssuch that p(c) ∈p(δ0). The component Γc of Γ \ ˜S containing c has nonemptyintersection with Γ1 because c is limiting poimt for Γ1.

As it was proved above, it

RAD ´O THEOREM AND ITS GENERALIZATION7follows that Γc is contained in Γ1. This contradiction (with c ∈Γ1) shows that Σ1is empty.Thus, we have proved that for each component Γ0 of Γ \ Σ the map f1 (equal tof on Γ0 and to 0 outside of it) is CR on the whole Γ.

As Γ contains no analyticdiscs, this f1 extends holomorphically into one-sided neighbourhouds of each pointof Γ. It follows from a boundary uniqueness theorem for holomorphic functions thatΓ \ Γ0 has zero Hausdorff3-measure, in particular, Γ0 is the single component ofΓ \ Σ.

In other words, the set Σ has zero Hausdorff(2n −1)-measure for general n,and its complement in Γ is locally connected (for every connected open set Γ′ ⊂Γthe set Γ′ \ Σ is also connected).References1. H. Alexander, Removable sets for CR-functions, Princeton Math.

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