On propagation of boundary continuity of holomorphic functions of several variables

On propagation of boundary continuity of holomorphic functions of several variables

SALLAFRANZEN AND´ BURGLINDJ ¨ORICKE

Abstract. We prove that continuity properties of bounded analytic functions in bounded smoothly bounded pseudoconvex domains in two-dimensional affine space are determined by their behaviour near the Shilov boundary. Namely, if the function has continuous extension to an open subset of the boundary containing the Shilov boundary it extends continuously to the whole boundary. If it ise.g.

H¨older continuous on such a boundary set, it is H¨older continuous on the closure of the domain. The statements may fail if the boundary is not smooth.

Mathematics Subject Classification (2000):32A40 (primary); 32E35 (secondary).

1.

Introduction

It is a classical observation that properties of analytic functions can be often directly read off from respective properties of their boundary values. The close relation be- tween functions and boundary values has played a big role in complex analysis and its applications, for instance in operator theory modeled on function spaces, scattering theory and others. One side of this relation, the idea to study objects by considering canonical extensions, has been successfully applied in many other situations, as harmonic extension of functions, extension of elements of Kleinian groups to hyperbolic space or a respective recent construction in conformal geom- etry. One of the aspects of the relation of properties of analytic functions and their boundary values is comparison of the respective continuity properties.

Investigation of problems of this kind was initiated by a classical theorem of Hardy and Littlewood about contour and solid H¨older continuity. Recall that a function f on a subsetE of

R

ⁿis H¨older continuous of orderα∈(0,1]if there is a constantC such that

sup

z,z∈E

|z−z|<δ

|f(z)− f(z)| ≤Cδ^αfor allδ >0. (1.1)

The smallest constantC for which (1.1) holds is called the H¨older seminorm of f. Received November 5, 2007; accepted in revised form January 29, 2008.

For a domain G in

C

^{n (n ≥} 1) we denote as usual by A(G)the algebra of analytic functions inGwhich extend continuously to the closureG. The following result of Hardy and Littlewood concerns the case whenGequals

D

, the unit disc in the complex plane [7].

Theorem HL. If f ∈ A(

D

⁾and its restriction to the unit circle is H¨older contin- uous of orderα,α ∈(0,1), then f is H¨older continuous of the same order in the closed disc

D

^.

It was proved later by Sewell [15] that actually the H¨older seminorms of the function on

D

, respectively on∂

D

, coincide.

In the sequel more general results on contour and solid continuity were proved.

Results were obtained for very general domains in the complex plane and arbi- trary quality of continuity instead of H¨older continuity. The quality of continuity is measured by the modulus of continuity. We call a continuous non-negative, non- decreasing functionµon the positive half-axis[0,+∞)a modulus of continuity if µ(0)=0 andµis subadditive,i.e.

µ(δ1+δ2)≤µ(δ1)+µ(δ2)forδ1≥0, δ2≥0.

Note that for any continuous function g on a compact convex set E ⊂

R

^{n the} functionµg(δ)=sup _z,z∈E

|z−z|<δ|g(z)−g(z)|forδ >0,µg(0)=0, is a modulus of continuity.

For any continuous function f on an arbitrary compact subset Eof

R

^{n there} exists a modulus of continuityµsuch that

sup

z,z∈E

|z−z|<δ

|f(z)− f(z)| ≤µ(δ)for allδ >0. (1.2)

Indeed, it is enough to extend f to a continuous function on a large closed ball and apply the preceding argument.

For unifying statements it will be often more convenient to use the follow- ing more general notion instead. Call a continuous non-negative, non-decreasing functionµon the positive half-axis[0,+∞)a generalized modulus of continuity if

µ(nδ)≤nµ(δ) (1.3)

for all natural numbersnandδ >0. We do not require here thatµ(0)=0. If sup

z,z∈E

|z−z|<δ

|f(z)− f(z)| ≤Cµ(δ)for allδ >0 (1.4)

for a generalized modulus of continuityµ, then we call f µ-continuous onE.

The more general point of view allowed to understand the underlying mecha- nism. It has a local and a global aspect. The local aspect is of potential theoretic

nature and based on the existence of local barrier functions. Local results hold for domains which are at a given point fat enough in a potential theoretic sense. The global aspect is based on the maximum principle for analytic functions.

The problem becomes more subtle for analytic functions of several variables.

Moreover, natural questions arise which do not exist in dimension one. Already the afore mentioned global aspect is more subtle for analytic functions in multi- dimensional domains. Indeed, for a bounded domainG ⊂

C

^n,n>1, any function in A(G) attains its maximum on the Shilov boundary S(G)of the domain (The definition of Shilov boundary will be recalled below). Notice that S(G)may be considerably smaller than the boundary ∂G of the domain. The question arises whether the continuity properties of functions in A(G)can be read off from those of their restrictions to a much smaller set than the boundary of the domain.

Moreover, the question arises whether one can weaken the a priori requirement f ∈ A(G). Will an analytic function inG be automatically in A(G)if it extends continuously to a smaller subset of the boundary? This question has no analogue in dimension 1. Here are the precise questions which emerge from a paper of Glicks- berg [5] although they are not asked there explicitly.

Question 1.1. For which bounded domainsG ⊂

C

ⁿ any continuous function on G∪ S(G)that is bounded and analytic onG extends to a continuous function on the closureG?

Question 1.2. For which bounded domainsG ⊂

C

ⁿ the previous question has an affirmative answer if S(G)is replaced by any relatively open subsetV of∂Gwith V ⊃ S(G)?

Glicksberg himself gave sufficient conditions on a domain to ensure a positive answer to the Question 1.1. Unfortunately, the conditions are difficult to verify and are only sufficient, which motivates the search for more geometric criteria. For a statement of his theorem we refer to the original paper [5].

In [9] it is proved that for regular Weil polyhedra in

C

^{n,n >} 1, Question 1.1 has a positive answer. Glicksberg’s conditions may not be satisfied for regular Weil polyhedra. Weil polyhedra are defined in the following way.

Let f_j, j = 1, . . . ,N be analytic functions in an open subsetU of

C

^{n. Sup-} pose the setG={^def z ∈ U : |f_j(z)| < 1,j = 1, . . . ,N}is connected and relatively compact inU. ThenGis called a Weil polyhedron. Weil polyhedra are pseudocon- vex domains. The polyhedron is regular if the level sets{|f_j| =1}of anyk-tuple (k ≤n) of different functions f_j intersect in general position. Note that polydiscs are regular Weil polyhedra. Moreover, all polynomially convex sets can be approx- imated from the outside by regular Weil polyhedra. The Shilov boundary of regular Weil polyhedra has a simple description (see [2]). It has real dimensionn.

In [9] it is also shown that for G being any regular Weil polyhedron and µ being any generalized modulus of continuity,µ-continuity onS(G)for a function as in Question 1.1 impliesµ-continuity of the function on the whole closureGofG.

On the other hand Glicksberg showed that for the “Hartogs triangle” G^def= {(z₁,z₂) ∈

C

^{2 : |z1| < |z2| < 1} ⊂}

C

² even the answer to Question 1.2 is neg- ative. The Hartogs triangle does not have smooth boundary. It turns out that for

domains in

C

²this is the only obstruction for a positive answer to Question 1.2. We prove here the following theorem. The formulation was proposed as a conjecture in [10].

Theorem 1.3. Let G ⊂

C

²be a bounded pseudoconvex domain with

C

^∞boundary.

Let V ⊂ ∂G be relatively open and contain the Shilov boundary S(G). Then for any continuous function f on G∪V that is bounded and analytic in G the following two statements hold.

1. f extends to a continuous function on the closure G.

2. If f isµ-continuous on V for a generalized modulus of continuityµthen f is µ-continuous on G. The respectiveµ-seminorms are related by a multiplicative constant depending only on G and V .

It follows that if in addition f is of class

C

^{∞ onG} ∪V then f extends to a

C

^∞ function on the closureG.

We do not know whether in Theorem 1.3 one can replace the neighbourhoodV of the Shilov boundary by the Shilov boundary itself. We do not know either in which way the latter would be related to the still open problem whether biholomor- phic mappings between bounded smoothly bounded pseudoconvex domains have smooth extension to the boundary.

ACKNOWLEDGEMENTS. The present paper was written while the second author was a guest of the Max-Planck-Institute.

2.

Proof of the theorem

We start with recalling some known results and notions. The first two statements extract the local and the global aspect of the relation of solid and boundary con- tinuity of analytic functions. They hold in arbitrary dimensions. We refrain from formulating the more subtle results that are known in dimension one and refer to the original literature (seee.g.[14]).

We start with the following global statement.

Lemma I. Let G be a bounded domain in

C

^{n and let f ∈ A(G)}. Then for every δ >0the supremum

sup{|f(z)− f(z)| :z,z∈G, |z−z|< δ}

is attained when one of the points z or zis contained in the boundary.

The lemma is well-known (see for instance [14]). For convenience of the reader we present the short proof.

Proof of the Lemma. Denotez=z+h. Both pointszandz+hare contained in G iffz∈G

(G−h). Fix the complex vectorhand apply the maximum principle to the analytic functionz → f(z+h)− f(z)inG

(G−h). Since the boundary of the setG

(G−h)is contained in the union of the boundaries∂G and∂(G−h), the supremum sup{|f(z+h)−f(z)| :z∈G

(G−h)}is attained if eitherz∈∂G orz∈∂(G−h). The second inclusion is equivalent toz=z+h∈∂G.

Here is a version of the local statement suitable also in the multi-dimensional set- ting. In the multi-dimensional setting it can be obtained from one-dimensional results by slicing. For an outline of the proof see [9].

Theorem I. Let G be a domain in

C

^{nwith z} ∈∂G. Assume that there is an open frustrum of a cone K with vertex at z lying in

C

ⁿ\G. Let µ be a generalized modulus of continuity. Then there exists a constant C > 0depending only on the aperture and height of the frustrum of the cone, such that for any function f ∈ A(G) the relation

sup

z∈∂G

|z−z|<δ

|f(z)− f(z)| ≤µ(δ)for anyδ >0

implies the relation sup

z∈G

|z−z|<δ

|f(z)− f(z)| ≤Cµ(δ)for anyδ >0.

The lemma and the theorem imply the statement on the relation between solid and boundary continuity for a wide class of multi-dimensional domains, in particular for those with

C

² boundary. Indeed, if a domain has

C

² boundary then for each of its boundary points the conditions of theorem I are satisfied and, moreover, the aperture and the height of the frustrum of the cones can be chosen to be independent of the boundary point.

Recall that the notion of the Shilov boundary was introduced in the general frame of Banach algebras. Here we consider only the case of the algebra A(G) for an arbitrary domain G ⊂

C

ⁿ. In this case the Shilov boundary S(G)is the intersection of all closed subsetsSofGfor which

|f(z)| ≤max

S |f|for allz∈Gand for all f ∈ A(G).

A setSwith the described property is called a boundary for A(G). Note that∂Gis a boundary, henceS(G)⊂∂G. MoreoverS(G)is a boundary itself,i.e.

|f(z)| ≤max

S(G)|f|for allz∈Gand for all f ∈ A(G).

For more detailed information we refer to the book [4].

Bremermann [2] initiated research on a geometric characterization of the Shilov boundary. He conjectured that the Shilov boundaryS(G)of a pseudoconvex domainGin

C

ⁿwith boundary of class

C

²is equal to the closure of the set₊(G) of strictly pseudoconvex boundary points. The first theorem in this direction was proved by Rossi.

Theorem II. [13]If G is a bounded pseudoconvex domain in

C

^nwith

C

^2boundary and G admits a Stein neighbourhood basis, then

S(G)=₊(G).

For one of the inclusions it is enough to require that the domain has

C

^2boundary.

Theorem III. [1] Let G be a bounded domain in

C

^{n with}

C

² boundary. Then the Shilov boundary is contained in the closure of the set of strictly pseudoconvex boundary points, i.e.

S(G)⊂₊(G).

Note that the domainGis not required to be pseudoconvex.

As for the other inclusion the condition of existence of a Stein neighbourhood basis ofGcan be removed, provided the boundary is of class

C

^{∞instead of}

C

^2. Theorem IV. [12] Let G ⊂

C

ⁿ be a bounded pseudoconvex domain with

C

^∞ boundary. Then

+(G)⊂ S(G).

Using the geometric notion of strict pseudoconvexity of boundary points instead of the notion of the Shilov boundary one can give a statement that holds for arbitrary bounded domains in

C

^{2 with}

C

³ boundary. Denote by ₋(G) the set of strictly pseudoconcave boundary points. Note that the set∂G\(₊(G)∪₋(G))is foliated into one-dimensional complex manifolds.

The following theorem holds.

Theorem 2.1. Let G ⊂

C

²be a bounded domain in

C

^2with

C

³boundary. Suppose V₊ ⊂ ∂G is relatively open in∂G and contains₊(G). Let f be any continuous function on G∪V₊which is bounded and analytic on G. Then

1. f extends to a continuous function on G.

2. If f isµ-continuous on G∪V₊for a generalized modulus of continuityµthen it isµ-continuous on G. Theµ-seminorms on G ∪ V₊ and on G differ by a multiplicative constant depending on G,V₊and the supremum norm of f on G.

In general, for non-pseudoconvex domainsG the Shilov boundary may be strictly smaller than₊(G). We do not know whether in Theorem 2.1 the set₊(G)may be replaced by the Shilov boundary itself.

The following lemmas relate Theorem 2.1 to Theorem 1.3.

Lemma 2.2. Let G be as in Theorem2.1and p ∈ ₋(G)\₊(G). Then there is a neighbourhood U_pof p in

C

²which depends only on G and p such that each bounded analytic function in G extends to an analytic function in U_p. The extension does not increase the supremum norm of the function.

Lemma 2.3. There exists a neighbourhood V of(G)^def=+(G)∪−(G)in∂G such that each function f as in Theorem2.1has continuous extension to G∪ V . Moreover, f isµ-continuous on V withµ-seminorm differing from that on V₊by a multiplicative constant depending on G, V and the supremum norm of f on G.

Proof. Indeed, choose for eachU_pas in Lemma 2.2 two open subsetsV_p andV_pof

∂G, p∈V_p

^Vp

^Up∩∂G. LetV^◦₊be an open subset of∂G,₊(G)⊂V^◦₊

V₊. Cover the compact set₋(G)\V^◦₊ by a finite number of V_p

j, j = 1. . . ,N. Denote byV the unionV = V^◦₊∪_N

j=1V_p

j. On each V_p

j the extension of f is Lipschitz continuous with Lipschitz seminorm depending on the supremum norm of f and the distance of V_pj to the complement of U_p. Since any generalized modulus of continuity exceeds const·δ, δ ≥ 0, for some positive constant const,

f is µ-continuous on each V_p

j. It is also µ-continuous on V₊ and V^◦₊

^V+. Notice, that for smallδ > 0 two pointszandz inV ⊂ ∂G of distance less than δfrom each other belong both to one of the setsV_p

j or toV₊. It follows that f is µ-continuous onV withµ-seminorm depending onG,V₊and the supremum norm of f onG.

Proof of Lemma2.2. By the conditions of the lemma, pis a minimal point of∂G in the sense that∂G contains no analytic curve that passes through p. For a small ballB_paround pthe neighbourhoodW_p =∂G∩B_pof pon∂Gis pseudoconcave from the side ofG. Denote the pseudoconvex set B_p\G byp. Let W_p

^Wp

be a smaller neighbourhood of pon∂G. By a theorem of Trepreau [16] (see also the more general result of Tumanov [17]) analytic functions on a neighbourhood of W_p have analytic extension to a one-sided neighbourhood

O

^{pof p(i.e. to one of} the connected components ofb_p\∂G for a small ballb_paround p). The one-sided neighbourhood

O

pdepends only onW_p and the extension preserves the supremum norm of the function. The set

O

pis contained inp. Indeed, take a vectorvso that for all sufficiently small positive numbersε the translatesW_p+εvare contained in p. Analytic functions inp extend to

O

p +εv for all smallε > 0, hence

O

p+εv⊂pfor all suchε. It follows that

O

p ⊂p, and thereforeG∪∂G∪

O

p

covers the ballb_p. Applying Trepreau’s theorem to a small translateW_p −εvinto G, we obtain the lemma.

Proof of Theorems1.3and2.1. To unify notation, put(G) = ₊(G)∪₋(G) in both theorems (in case of Theorem 1.3₋(G)is empty). We may assume that there is a neighbourhood V of (G)in ∂G such that f is bounded and analytic inG, continuous inG ∪V andµ-continuous onV for some generalized modulus of continuityµ. (For Theorem 2.1 the present generalized modulus of continuity differs from the one in the statement of Theorem 2.1 by a multiplicative constant that depends onG,V and the supremum norm of f). Choose a neighbourhoodV^◦ (in∂G) of(G),V^◦

^V.

The following lemma is a local statement. It implies immediately continuous extension of the function f to the set∂G\V^◦. Successive application of the lemma allows to increase the quality of continuity of the function on this set (see Corol- lary 2.5).

Lemma 2.4. Let G,V and V be as above. For each point p^◦ ∈∂G\V there are a^◦ small ball b_paround p, a positive numberτp≤1and a constant C_p>1such that the following holds.

Suppose f andµare as above andν ≥µis another generalized modulus of continuity. Suppose

sup

z,z∈G

|z−z|<δ

|f(z)− f(z)| ≤ν(δ)for allδ >0. (2.1)

Then

sup

z,z∈b_p∩G

|z−z|<δ

|f(z)− f(z)| ≤C_pν(δ)^1−τp µ(δ)^τpfor allδ >0. (2.2)

The constantsτpandC_pdepend only onp,G,V^◦ andV but not onνandµ. Note that any function as in the beginning of the proof satisfies (2.1) with ν(δ) ≡ c sup_G|f| for any constant c ≥ 2. Redefiningµ(δ) to be constant for δ >di am(G)the constantcmay be chosen so thatν≥µ.

We postpone the proof of the lemma. The lemma has the following corollary.

Corollary 2.5. Suppose f is as above and satisfies(2.1)for a functionνas in the lemma. Then f has continuous extension to ∂G. Moreover, there exist a positive constantτ ≤1and a constant C >1(depending only on G and V)such that for all positiveδ

sup

z,z∈G,|z−z|<δ |f(z)− f(z)| ≤Cν(δ)^1−τµ(δ)^τ. (2.3) Proof of the Corollary. Inequality (2.2) implies that for each p ∈∂G\V^◦ the func- tion f extends continuously tob_p∩G, being uniformly continuous on the dense open subsetb_p∩G. Since f extends continuously toV ⊃V^◦, continuous extension to the whole boundary∂Gfollows.

PutV_p^def=b_p∩∂Gand choose an open subsetV^◦_pof∂G,p∈V^◦_p

^Vp. Cover the compact set∂G\V^◦ by finitely many of theV^◦_p. Letτ be the smallest of the respective constants τp. Notice, that for smallδ > 0 two points z andz in∂G belong both to one of the sets V_p or to V if|z−z| < δ. Sinceµ(δ) ≤ ν(δ)for δ≥0, andτ ≤τp, the inequality

sup

z,z∈∂G

|z−z|<δ

|f(z)− f(z)| ≤maxC_p· ν(δ)^1−τ µ(δ)^τ (2.4)

holds forsmallδ >0. Increasing the constant, we get (2.4) forarbitraryδ >0. The- orem I and Lemma I imply (2.3) after further increasing the multiplicative constant.

The corollary is proved.

We will now finish the proof of the theorems by successively applying the corollary.

End of proof of the theorems. Let f be as in the beginning of the proof of the the- orems. Putµ0(δ)≡ Aforδ ≥0, where A= c sup_G|f|for a suitable constantc exceeding 2 so thatµ0 ≥µ. LetC andτbe the constants of the corollary. Apply- ing the corollary withν =µ0we obtain that f extends to a continuous function in Gand

sup

z,z∈G

|z−z|<δ

|f(z)− f(z)| ≤C A^1−τµ(δ)^τ for allδ >0.

Putµ1(δ)=C A^1−τµ(δ)^τ =Cµ0(δ)^1−τµ(δ)^τ forδ≥0. Thenµ1is non-negative, non-decreasing, satisfies condition (1.3) and is not smaller thanµ. We may apply the corollary with ν = µ1. Repeated application of the corollary gives for any natural number j the following estimate

sup

z,z∈G

|z−z|<δ

|f(z)− f(z)| ≤µj(δ)for allδ >0, (2.5)

where

µj(δ)=C^{l=0j−1(1−τ)l}A^(1−τ)jµ(δ)^{τl=0j−1(1−τ)l}, δ >0. (2.6) Indeed, suppose (2.5) is true for j−1 with the non-negative, non-decreasing func- tionµj−1defined by (2.6). Suppose thatµj−1≥µandµj−1satisfies (1.3). Then the corollary gives

sup

z,z∈G

|z−z|<δ

|f(z)− f(z)| ≤C

µj−1(δ)_1−τ

µ(δ)^τ for allδ >0.

Denote the function on the right hand side by µj. It is again non-negative, non- decreasing, not smaller thanµand satisfies (1.3), since both functionsµj−1 and µhave this property. Moreover, (2.6) holds forµj. Indeed, since (2.6) is true for

j−1,

µj(δ)=C

C^{l=0j−2(1−τ)l}A^{(1−τ)j−1}µ(δ)^{τl=0j−2(1−τ)l}1−τ

µ(δ)^τ

=C^{l=0j−1(1−τ)l}A^(1−τ)jµ(δ)^{τl=0j−1(1−τ)l}forδ≥0, (2.7) what we needed to prove.

Note that lim

j→∞µj(δ)=C^{1−(1−τ)1} µ(δ)^{τ1−(1−τ)1} =C^1τµ(δ)forδ≥0.

Since (2.5) is true for each j, we obtain the theorem with constantC^τ1. HereC and τ are the constants in the corollary. The theorems are proved.

It remains to prove the lemma.

Proof of Lemma2.4. LetW be a connected component of∂G\(G).W is a Levi- flat hypersurface and hence it is foliated by1-dimensional complex manifolds which are embedded or injectively immersed intoW. Recall thatV^◦ is an open subset of

∂G,(G)⊂V^◦ ⊂ V^◦ ⊂V.

Take a point p∈W\V^◦. Let

L

pbe the Levi-leaf ofW throughp.

There is a curveγp in

L

^{p,γp : [0,1] →}

L

^p, which joins pwith a point in W ∩V^◦. Otherwise

L

^pwould be contained in the compact subsetW\V^◦ ofW and the same would apply to its closure

L

¯p (closure with respect to Euclidean topol- ogy in

C

²). As a consequence,

L

¯p would be a compact set which is the union of injectively immersed complex one-dimensional analytic manifolds (see also [11]), which is impossible [6]. Removing parts ofγpbetween self-intersection points and approximating, we may assume that γp is real analytic on a slightly larger inter- val, in particular it extends to an analytic diffeomorphism from a neighbourhood of I_ε = [−ε,1]in

C

to a neighbourhood ofγp(I_ε)on

L

^p. We may keep the condition γp(0) = p. γp(I_ε)can be considered as the symmetry axis of a rectangle in the complex leaf

L

p. Our goal is now to foliate a neighbourhood ofγp(I_ε)in G by holomorphic rectangles close to the mentioned one. More precisely, we will de- fine a diffeomorphic map

H

⁽⁼

H

p)¹of class

C

¹from a neighbourhood of a closed rectangular box

Q(=Q_ε,σ)={(z₁,z₂):Rez₁∈I_ε,Imz₁∈[−ε, ε],Rez₂∈[−σ, σ],Imz₂∈[0, σ]}

onto a subset ofGsatisfying the following properties:

(1) Denote by R_z2, z₂ ∈ [−σ, σ] ×i[0, σ], the rectangle obtained from Q by fixingz₂. The mapping

H

is holomorphic on eachR_z2.

(2) The “lower” face of

H

(Q),

H

(Q∩ {Imz₂ = 0}), is contained inW ⊂ ∂G.

Each conformal rectangle

H

(R_x2), x₂ ∈ [−σ, σ], is contained in one of the leaves ofW. In particular,

H

(R₀)containsγ (I_ε).

This is obtained in the following way. LetX andY be real

C

²vector fields defined at points of

L

^{p close toγ (I}ε). Let

F

^Y,y,

F

^X,x be their flows. We may choose the vector fields so that

H

^{(x+i y,0)def=}

F

^Y,y◦

F

^X,x(p)coincides with the analytic extension of γp for x,y in a neighbourhood of I_ε × [−ε, ε]. Extend X,Y to

C

² real vector fields in a neighbourhood ofγ (I_ε)on∂Gwhich are complex tangent to

∂G at each point. Letv : [−σ, σ] → W be a

C

²curve which is transverse to the Levi leaves, withv(0) = p. Put H˜(x +i y,t) =

F

Y,y ◦

F

X,x ◦v(t). For fixedt the mapping

H

˜ ^{maps Rt} into the Levi-leaf throughv(t). The mapping may not be holomorphic inz =x+i y, but it is

C

¹close to the holomorphic mapz→

H

^(z,0).

1All further constructions depend on the pointp, but we will skip the indexp.

There is a

C

¹ family of reparametrizations ϕt (each ϕt being a map from a neighbourhood ofI_εto a neighbourhood ofI_ε) such thatϕ_◦=id and

H

(z,t)= ˜

H

(ϕt(z),t)

is holomorphic for eacht. Indeed, denote by

H

˜j the coordinate functions of

H

˜^.ϕt

must satisfy the equations

∂(¯

H

˜j(ϕt,t))=∂

H

˜j(ϕt,t)∂ϕ¯ t + ¯∂

H

˜j(ϕt,t)∂ϕt =0, j =1,2. (2.8) Each of the two equations (2.8) gives a well defined non-linear∂¯-operation forϕt

on the set where the respective∂

H

˜j does not vanish. Since for fixedt,

H

˜(z,t)= (

H

˜1(z,t),

H

˜2(z,t)) maps into a Levi-leaf, there is an analytic function F_t with non-vanishing gradient such that

F_t(

H

˜^1(z,t),

H

^{˜2(z,t))=0 (2.9)} for z in a neighbourhood of I_ε. Taking ∂- and∂¯-derivatives of (2.9) and taking into account that

H

˜(·,t)is

C

¹close to an analytic diffeomorphism we obtain that if

∂jF_t =0 then∂

H

˜i =0 for the other indexi and at the set where both derivatives

∂1F_t and∂2F_t do not vanish, ^∂_∂^¯H_H¹

1 = ^∂H_∂H^{¯ 2}₂. This gives a well-defined∂¯-equation for ϕt in a neighbourhood ofI_εwhich has a solution close to the identity. Moreover, the solutionϕtdepends

C

^1ontbecause

H

^˜ is of class

C

². The mapping

H

is constructed on the lower faceQ∩ {Imz₂=0}.

To define

H

on the whole Q we denote forz ∈ ∂G byn(z)the unit inner normal to ∂G. This is a

C

² vector field on ∂G. Approximaten on the compact setγ ([−ε,1])by an analytic vector fieldN in a neighbourhood (in

C

²) of this set ( [8]). Put

H

^{(z,t+i s)=}

F

^N,s(

H

^(z,t)), s ∈ [0, σ]for a small positive numberσ.

SinceN is holomorphic in a neighbourhood ofγ (I_ε)and

H

(z,t)is holomor- phic inz, the function

H

(z,t +i s)is holomorphic inz. The construction of

H

^on Q is completed.

Decreasingε > 0 andσ > 0 if necessary we may assume that the “right”

“lower” 2-face of Q_ε,σ, Q_ε,σ ∩ {Rez₁ = 1,Imz₂ = 0}, is mapped intoV^◦\(G). Denote the “right” faceQ_ε,σ ∩ {Rez₁=1}byQ^r_ε,σ.

The following lemma states that at points of

H

(Q^r_ε,σ)the function has the de- sired continuity properties. The proof uses the local Theorem I and a pluriharmonic measure estimate.

Lemma 2.6. The estimate sup

p,p∈H(Q^r_ε,σ)

|p−p|<δ

|f(p)− f(p)| ≤Cν(δ)^1−τµ(δ)^τ= ˜^defµ(δ)

holds for a constantτ∈(0,1)and a constant C.

Proof. Let p,p ∈ G be two points in a sufficiently small neighbourhood of

H

(Q^r_ε,σ),|p− p|< δfor a smallδ >0. Suppose first that one of the points, say p, has distance to∂Gnot exceedingδ. Then, since

H

(Q^r_ε,σ∩ {Imz₂ =0})⊂ V,^◦ there is a point p^∗ ∈V^◦ such that|p− p^∗| ≤δ. By the local Theorem I,

|f(p)− f(p^∗)| ≤constµ(δ).

The same inequality withδ replaced by 2δ holds for p. Triangle inequality and properties ofµyield

|f(p)− f(p)| ≤Cµ(δ) (2.10) for this case.

Let now p,p ∈ G be contained in a neighbourhood

N

^of

H

(Q^r_ε,σ),|p− p|< δand both have distance bigger thanδto the boundary∂G. The function

q→ f(q)− f(p− p+q) (2.11) is analytic in the intersection

N

^{∩ {dist(q, ∂G) > δ}}, satisfies (2.10) on its bound- ary part contained in {dist(q, ∂G) = δ}and has supremum norm not exceeding ν(δ). Pluriharmonic measure of the latter boundary part with respect to

N

^∩ {dist(q, ∂G) > δ}computed at points of

H

(Q^r_ε,σ)∩ {dist(q, ∂G) > δ} has an estimate from below by a positive constantτ. (This can be seen by slicing with complex lines). Hence, on

H

(Q^r_ε,σ)∩ {q ∈ G : dist(q, ∂G) > δ} the absolute value of the function (2.11) does not exceed

Cµ(δ)_τ

ν(δ)^1−τ.

The value of the function (2.11) at the pointq = p ∈

H

(Q^r_ε,σ)∩{dist(q, ∂G) > δ}

equals f(p)− f(p). This proves the required inequality in the second case.

Lemma 2.6 is proved.

End of proof of Lemma 2.4. Let ε ∈ (0, ε) and consider the rectangular box Q_ε,σ obtained fromQ_ε,δby shrinking in thez₁-direction. We will prove that esti- mate (2.2) of the lemma holds on the set

H

(Q⁰_ε,σ), whereQ⁰_ε,σ is the intersection of the interior of Q_ε,σ with{|Rez₁|< ε}in

C

^2,

Q⁰_ε,σ^def={(z₁,z₂):z₁∈(−ε, ε)+i(ε, ε),z₂∈(−σ, σ )+i(0, σ )}.

The set

H

(Q⁰_ε,σ)is contained inG and contains the intersection ofG with a ball around p. (Recall, that

H

(0,0)= p.)

The diffeomorphism

H

and its inverse

H

⁻¹ preserve distance up to a multi- plicative constant,i.e.

|

H

⁻¹(p)−

H

⁻¹(p)| ≤c|p−p|, p,p∈

H

(Q_ε,σ),

|

H

^(z)−

H

^{(z)| ≤c|z−z|, z,z ∈Q}ε,σ.

Let p,p ∈

H

(Q⁰_ε,σ),|p− p| < δ withδ > 0. Write

H

⁻¹(p)= (z₁,z₂) ∈ Q_ε,σ,

H

^−1(p)=(z1,z₂)∈Q_ε,σ and consider the function

z→(f ◦

H

)(z,z₂)−(f ◦

H

)(z+(z₁−z₁),z₂). (2.12) Ifδ >0 is small, this function is inA(R)for the rectangleR= [−ε,1] +i[−ε, ε]. (Note that Imz₂>0, Imz₂ >0). The value of the function (2.12) atz₁equals

(f ◦

H

)(z₁,z₂)−(f ◦

H

)(z₁,z₂)= f(p)− f(p).

The supremum norm of the function (2.12) does not exceedν(ccδ). On the right side{1} ×i[−ε, ε]of the rectangle Rits absolute value does not exceedµ(˜ ccδ). Therefore on the relatively compact part{(−ε, ε)+i(−ε, ε)}of R, in particular at the pointz₁, its absolute value does not exceed

C_pν(δ)^1−τpµ(δ)^τp

for suitable constantsτp ∈(0,1]andC_p > 0. Here we used property (1.3) ofµ andν, the fact thatν ≥µ, the definition ofµ˜ in Lemma 2.6 and harmonic measure estimate onR.

Lemma 2.4, and hence theorems 1.3 and 2.1, are proved.

We do not know whether in theorems 1.3 and 2.1 one needs to require that the function f is bounded inG.

Under some additional hypothesis this condition can be removed. The follow- ing lemma describes sufficient conditions in the case of pseudoconvex domains.

Lemma 2.7. Let G ⊂

C

²be a bounded pseudoconvex domain with

C

^2boundary.

Suppose each set of an exhausting sequence of compact subsets of∂G\+(G)has a Stein neighbourhood basis. Then any analytic function in G that extends contin- uously to a neighbourhood of₊(G)in∂G is bounded in G.

Proof. LetVbe a neighbourhood of₊(G)in∂Gand f a continuous function on G∪ V that is holomorphic inG. LetV^◦ be open in∂G and₊(G) ⊂ V^◦

^V.

There is a compact subset K ⊂G∪V which contains a one-sided neighbourhood of each point of V^◦ (i.e. the intersection ofG with a small ball around the point).

The function f is bounded onK.

The set∂G\V^◦ is a compact subset of∂G\₊(G). It is contained in a compact setκ⊂∂G\₊(G)which has a Stein neighbourhood basis. Since any pseudocon- vex domain can be exhausted by smoothly bounded strictly pseudoconvex domains, there exists a neighbourhood basisU_nofκconsisting of smoothly bounded strictly pseudoconvex domains. ConsiderG\U_n and smoothen its boundary part which is contained in a small neighbourhood of∂U_n∩∂G ⊂ V^◦. We may assume that the changed boundary part is contained in K and the thus obtained domain G_n has

C

^2boundary.

The function f is continuous onG_n ⊂G∪V. The part of the boundary∂G_n which is outside K is strictly pseudoconcave hence does not meet ₊(G_n). By Theorem III

max

Gn

|f| ≤max

K |f|.

This is true for allnand theG_nexhaustG. The lemma is proved.

Necessary and sufficient conditions for the existence of a Stein neighbourhood basis of a Levi-flat hypersurface are not known. See [3] for an account.

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Department of Mathematics Stockholm University SE-106 91 Stockholm, Sweden [email protected]

Max-Planck-Institut f¨ur Mathematik P.O. Box: 7280 53072 Bonn, Germany and

Department of Mathematics Stockholm University SE-106 91 Stockholm, Sweden [email protected]