ON BOUNDARY PROPERTIES OF ASYMPTOTICALLY HOLOMORPHIC FUNCTIONS Текст научной статьи по специальности «Математика»

ISSN 2074-1871 Уфимский математический журнал. Том 14. № 3 (2022). С. 131-144.

ON BOUNDARY PROPERTIES OF ASYMPTOTICALLY HOLOMORPHIC FUNCTIONS

A. SUKHOV

Abstract. It is well known that for a generic almost complex structure on an almost complex manifold (M, J) all holomorphic (even locally) functions are constants. For this reason the analysis on almost complex manifolds concerns the classes of functions which satisfy the Cauchv-Riemann equations only approximately. The choice of such a condition depends on a considered problem. For example, in the study of zero sets of functions the quasiconformal type conditions are very natural. This was confirmed by the famous work of S. Donaldson. In order to study the boundary properties of classes of functions (on a manifold with boundary) other type of conditions are suitable. In the present paper we prove a Fatou type theorem for bounded functions with d j differential of a controled growth on smoothly bounded domains in an almost complex manifold. The obtained result is new even in the case of Cn with the standard complex structure. Furthermore, in the case of Cn we obtain results with optimal regularity assumptions. This generalizes several known results.

Keywords: almost complex manifold, 9-operator, admissible region, Fatou theorem. Mathematics Subject Classification: 32H02, 53C15

1. Introduction

This paper is a continuation of work [17]. We improve the main results of [17] establishing a general version of the Chirka-Lindelof principle and the Fatou type theorem for bounded asymptotically holomorphic functions on almost complex manifolds. These functions admit the antiholomorphic part of the differential satisfying some asymptotic growth conditions near the boundary. Such classes of functions naturally appear in Several Complex Variables, PDE and related topics. Our results extend the known results [2], [5], [8], [14], [15] obtained for the case of Cn with the standard complex structure. Note that in this case our results also are knew. Moreover, we obtain the results in Cn with optimal regularity assumptions.

2. Almost complex manifolds and almost holomorphic functions

This is a preliminary section. We recall basic notions of the almost complex geometry making the presentation of our results more convenient. Throughout the paper we assume that manifolds and almost complex structures are of class C^ (the word «smooth» means the regularity of this class). However, our main results are also valid under considerably weaker regularity assumptions.

2.1. Almost complex manifolds. Let M be a smooth manifold of real dimension 2n. An almost complex structure J on M is a smooth map which associates to every point p E Ma linear isomorphism J(p) : TPM ^ TPM of the tangent space TPM such that J(p)2 = — Ip; here Ip denotes the identity map of TPM. Thus, every linear operator J(p) is a complex structure

A. Sukhov, On boundary properties of asymptotically holomorphic functions.

© A. Sukhov 2022.

Submitted April 28, 2022.

on a vector space TPM in the usual sense of Linear Algebra, When J is fixed, a couple (M, J) is called an almost complex manifold of complex dimension n,

A fundamental example of an almost complex structure is given by the standard complex structure Jst = JÍ¿2) on M = R2, This linear operator is represented in the canonical coordinates of R2 bv the matrix

/0 -1\

Jst = [x o) (2-D

More generally, the standard complex structure Jst on R2n is represented by the block diagonal matrix diag(J{¡2,..., J^) (usually we drop the notation of dimension because its value will be clear from the context). Setting iv := Jstv for a vector v G R2n, we identify the real space (R2n, Jst) with the complex linear space Cn; we use the notation z = x + iy = x + Jsty for the standard complex coordinates z = (z1,..., zn) G Cn,

Let (M, J) and (M', J') be smooth almost complex manifolds, A C^map f : M' M M is called (J, J)-complex or (J', J)-holomorphic if it satisfies the Cauchy-Riemann equations

df o J' = J o df. (2.2)

In particular a map f : Cn ^ Cm is (Jst, Jsi)-holomorphic if and only if each component of

Every almost complex manifold (M, J) can be viewed locally as the unit ball B in Cn equipped with a small (in any Cm-norm) almost complex deformation of Jst. The following well-known statement is often useful.

Lemma 2.1. Let ( M, J) he an almost complex manifold. Then for every point p G M, every m ^ 0 and A0 > 0 there exist a neighborhood U of p and a coordinate diffeomorphism z : U ^ B such that z(p) = 0 dz (p) o J (p) o dz-1(0) = Jst, and the direct i mage z*(J) := dz o J o dz-1 satisfies ||z*(J) - JstMc^iB ^ Ao-

Proof There exists a diffeomorphism z from a neighborhood U' of p G M onto B satisfying z(p) = 0 after an additional linear change of coordinates one can achieve dz(p)oj(p)odz-1(0) = Jst (this is a classical fact from the Linear Algebra), For A > 0 we consider the isotropic dilation h\ : t M- A-1t in R2n and the composition = h\oz. Then limA^0 ||(z\)*(J) -Jstllc™(S) = 0 for every m ^ 0 Setting U = z-1(B) for A > 0 small enough, we obtain the desired statement. In what follows we often denote the structure z*( J) again by J viewing it as a local representation of J in the coordinate svstem (z). □

Recall that an almost complex structure J is called integra ble if (M, J) is locally biholomor-phie in a neighborhood of each point to an open subset of (Cn, Jst). In the case of complex dimension 1 every almost complex structure is integrable. In the case of complex dimension > 1 integrable almost complex structures form a highly special subclass in the space of all almost complex structures on M; an efficient criterion of integrablitv is provided by the classical theorem of Newlander-Nirenberg [9],

2.2. Pseudoholomorphic discs. Let (M, J) be an almost complex manifold of dimension n > 1. For a "generic" choice of an almost complex structure, any holomorphie (even locally) function on M is constant because the Cauchy-Riemann equations are overdetermined. For the same reason M does not admit non-trivial J-complex submanifolds of complex dimension > 1. The unique exceptional case arises when J-complex submanifolds are of complex dimension 1, They always exist at least locally,

where M' has the complex dimension 1. These holomorphie maps are called J-complex (or J-holomorphie or pseudoholomorphic ) curves. Note that we consider here the curves as maps

i.e. we consider parametrized curves. We use the notation D = {( E C : |(| < 1} for the unit disc in C always assuming that it is equipped with the standard complex structure Jst. If in the equations (2,2) we have M' = D, we call such a map f a J-complex disc or a pseudoholomorphic disc or just a holomorphie disc when the structure J is fixed,

A fundamental fact is that pseudoholomorphic discs always exist in a suitable neighborhood of any point of M; this is the classical Nijenhuis-Woolf theorem (see [10]), Here it is convenient to rewrite the equations (2,2) in local coordinates similarly to the complex version of the usual Cauchy-Riemann equations.

Everything will be local, so (as above) we are in a neighborhood Q of 0 in Cn with the standard complex coordinates z = (z1,..., zn). We assume that J is an almost complex structure defined on Q and J(0) = Jst. Let

z: D M Q, z:(m z(()

be a J-complex disc. Set ting ( = £ + ir] we write (2,2) in the form zv = J (z) This equation can be written as

— A(z)z^ = 0, ( E D. (2.3)

Here a smooth map A : Q M Mat(n, C) is defined by the identity L(z)v = Av for any vector v E C^d L is an R-linear map defined by L = (Jst + J)-1(Jst — J). It is easy to check that the condition J2 = — Id is equivalent to the fact that L is C-linear, The matrix A(z) is called the complex matrix of J in the local coordinates z, Locally the correspondence between A and J is one-to-one. Note that the condition J(0) = Jst means that A(0) = 0, If t are other local coordinates and A' is the corresponding complex matrix of J in the

A =(t ZA + t* )(tz + tzA)-1 (2.4)

Note that one can view the equations (2.3) as a quasilinear analog of the Beltrami equation for vector-functions. From this point of view, the theory of pseudoholomorphic curves is an analog of the theory of quasi-conformal mappings.

= i r

2n% Jd u — C

This classical linear integral operator has the following properties (see [19]):

(i) T : Cr (D) ^ Cr+1 (D) is a bounded linear operator for every non-integer r > 0 (a similar property holds in the Sobolev scales, see below). Here we use the usual Holder norm on the space Cr (D),

(ii) (Tf )-£ = f i.e. T solves the 5-equation in the unit disc,

(iii) the function Tf is holomorphie on C \ D,

Fix a real non-integer r > 1 Let z : D ^ Cra, z : D 9 ( M- z(() be a J-complex disc. Since the operator

: z —> w = z — TA(z)z^

maps the space Cr(D) into itself, we can write equation (2,2) in the form (tyj(z))-^ = 0, Thus, the disc z is J-holomorphic if and only if the map ^j(z) : D —> C^s Jsi-holomorphic, A

function theorem the operator is invertible in the space Cr (D) and we obtain a bijeetive correspondence between J-holomorphic discs and usual holomorphie discs. This implies easily the existence of a J-holomorphic disc in a given tangent direction through a given point of M, as well as a smooth dependence of such a disc on a deformation of a point or a tangent vector, or on an almost complex structure; this also establishes the interior elliptic regularity of discs. This is the classical Nijenhuis-Woolf theorem, see [10],

Let ( M, J) be an almost complex manifold and E c M be a real submanifold of M. Suppose that a J-complex disc f : D ^ M is continuous on D, With some abuse of terminology, we also call the image /(D) simply by a disc and we call the image f(6D) the boundary of a disc. If

( D) c E E

that f is attached to E, If 7 c bD is an arc and f(7) c E, we say that f is glued or attached E

2.3. The ^-operator on an almost complex manifold (M,J). Now we consider the second special class (together with pseudoholomorphic curves) of holomorphic maps. Consider first the situation when J be an almost complex structure defined in a domain Q c Cn; one

J M

A C function F : Q ^ C is (J, Jst )-holomorphie if and only if it satisfies the Cauchy-Riemann equations

F-z + FzA(z) = 0, (2.6)

where Fz = (dF/dz1,..., dF/dzn) and Fz = (dF/dz1,..., dF/dzn) are regarded as row-vectors. Indeed, F is (J, Jst) holomorphic if and only if for every J-holomorphic disc z : D ^ Q the composition F o z is a usual holomorphie function that is d(F o z)/d( = 0 on D, Then the Chain rule in combination with (2,3) leads to (2,6), Generally the only solutions to (2,6) are

J A

Note also that (2,6) is a linear PDE system while (2,3) is a quasilinear PDE system for a vector D

Every 1-differential form 0 on (M, J) admits a unique decomposition 0 = 01,0 + 00,1 with respect to J. In particular, if F : (M, J) ^ C is a C^complex function, we have dF = dF1,0 + dF0,1, We use the notation

djF = dF10 and djF = dF0,1. (2.7)

In order to write these operators explieitelv in local coordinates, we find a local basis in the space of (1,0) and (0,1) forms. We view dz = (dz]_,..., dzn)f' and dz = (dz1,..., dzn)f' as vector-columns. Then the forms

a = (a1,..., an) = dz — Adz and a = dz — Adz (2.8)

form a basis in the space of (1,0) and (0,1) forms respectively. Indeed, it suffices to observe that a 1-form ft is of type (1,0) (resp, (0,1)) if and only if for every J-holomorphic disc z the pull-back z*P is a usual (1,0) (resp, (0,1)) form on D, Using equations (2,3), we obtain the claim.

Now we decompose the differential dF = Fzdz + Fzdz = dJF + dJF with respect to the basis a, a using (2,8), We obtain the explicit expression

djF = (Fz (I — AA)-1 + Fz (I — AA)-1A)a (2.9)

It is easy to check that the holomorphv condition dJF = 0 is equivalent to (2.6) because (I — AA)-1A(I — AA) = A. Thus, "

djF =(Fz +FzA)(I — ^A)-la

We note that the matrix factor (I — AA)-1 as well as the forms a affect only the non-essential constants in local estimates of the d^operator near a boundary point which we will perfom in the next sections. So the reader can assume that this operator is simply given by the left hand expression of (2.6).

F C1 Q

almost complex 'manifold (M, J) of dimension n. We call F a subsolution of the dJ operator

or simply a dj-subsolution on Q if there exists constants C > 0 and r > 0 such, that

|| djF(z) |K Cdist(z, bQ)-1/2+T (2.10)

for all z E Q. Here we use the norm with respect to any fixed Riemannian metric on M.

Obviously, non-constant 5j-subsolutions exist in a sufficiently small neighborhhod of any point of M. For example, each function F of class C1 in an open neighborhhod of the compact set Q is a 5j-subsolution on Q. Of course, any C1 function F with uniformly bounded djF on Q, satisfies (2,10), This subclass of functions was studied in [17]. In the case of Cra, a similar class of functions appeared in [5].

Let F be a Sj-subsolution on Q, Suppose that A is the complex matrix of J in a local chart U and z : D M U is a J-complex disc. It follows by the Chain Rule and (2.3) that

( F o z)s = (F-z + FZA)z-t.

Thus, if h : D M Q is a J-complex disc of class C 1(D^, then the composition F o h has a S-derivative satisfying (2.10) on D that is F o h is a 5jsi-subsolution on D. Note that the constant C and t appearing in the upper bound of type (2.10) for the d(F o h) depend only on constants from the upper bound on djF in (2.10), and the C1 norm of h on D as well. In particular, if (ht) is a family of J-complex discs in Q and C 1-norms of these discs are uniformly bounded with respect to t, then then one can find C > 0 and r > 0 independent of t for the upper bound of || dj(F o ht) ||.

2.4. One-dimensional case. Recall some boundary properties of subsolutions of the 5-operator in the unit disc.

Denote by Wk'p(D) the usual Sobolev classes of functions admitting generalized partial derivatives up to the order k in LP(D) (in fact we need only the case k = 0 and k =1). In particular W°'P(D) = Lp(D), We will always assume that p > 2.

Denote also by || f ||^= supD |/| the usu al sup-norm on the space L^(D) of complex functions D

Lemma 2.2. Let f E L~(D) and f-^ E LP(D) for some p> 2. Then

(a) f admits a non-tangential limit at almost every point ( E 6D.

(b) if f admits a limit along a curve in D approaching 6D non-tangentially at a boundary point e%e E bD, then f admits a non-tangential limit at e%e.

(c) for each positive r < 1 there exists a constant C = C(r) > 0 (independent of f) such, that for every (j E rD j = 1, 2 one has

If(C1) — /«2)| ^ C(| f ||« + || f-c ||LP(D))|C1 — C2|1-2/p (2.11)

The proof is contained in [17].

D > 0

g E L™(pD) and g^ E Lp(pD), The fonction f(() := g(p() satisfies the assumptions of Lemma 2.2 on D. Let 0 < a < p and let |^ | < a, j = 1, 2. Set (j = ^/p. Then |Q | < r = a/p < 1, j = 1, 2. Applying (c) Lemma 2.2 to f we obtain:

|0(n) — g(T2)| ^ ( C(r)/p1-2/p)(|| gU +p || rc H^rD))|n — T2|1-2/p (2.12)

Note that the constant C = C(r) = C(a/p) depends only on the quotient r = a/p < 1, If r is separated from 1, the value of C is fixed.

3. Main results

First we introduce an almost complex analog of an admissible approach which is classical in

Cn

Let Q be a smoothly bounded domain in an almost complex manifold (M, J). Notice that any domain with boundary of class C2 satisfies all assumptions imposed below. Fix a hermitian M J

changes only constant factors in estimates. We measure all distances and norms with respect to the choosen metric.

Let p E bQ be a boundary point, A non-tangential approach to bQ at p can be defined as the limit along the sets

Ca(p) = {q E Q : dist(g,p) < a5p(q)}, a > 1. (3,1)

Here 5P(q) denotes the minimum of distances from q to the tangent plane Tp(bQ) and to bQ.

Q

the limit along the sets

Aa,s(p) = {qE Q: dp(q) < (1 + a)5p(q), dist(p,q)2 < a(q)}, a> 0, e > 0. (3.2) Here dp (q) denotes the distance from q to the holomorphic tangent space

Hp(bQ) = Tp(bQ) n JTp(bQ). C n Q

normal direction and can be tangent in the directions of the holomorphic tangent space. Definition 3.1. A function F : Q ^ C has an admissible limit L at p E bQ if

lim F(q) = L for all a,e > 0.

Aa,s(p)Bq

Next we need the following notion,

Q ( M, J)

of complex dimension n and p E bQ be a boundary point. A real curve 7 : [0,Q of class C 1([0,1]) is called an admissible p-curve if 7(1) = p and 7 is transverse to the tangent space Tp(bQ) (i.e. the tangent vector of 7 at p is not contained in Tp(bQ)).

F Q L E C

7 if there exists limi^1(F o 7)(t) = L.

Our first main result is the following analog of the Chirka-Lindelof principle [2].

Q ( M, J)

of complex dimension n. Suppose that a complex function F E Lr(Q) is a dJ-subsolution

Q F

E Q F

A similar result is obtained in [17] under considerably stronger assumptions. First, the Q

7

there by the stronger condition of boundedness of dJF(z) on Q.

We use the notation f(x) ~ g(x) for two functions f(x), g(x) when there exists a constant C > 0 such that C-1g(x) ^ f(x) ^ Cg(x). In what follows the value of constants C can change from line to line.

As an application of the Chirka-Lindelof principle we establish Fatou type results for d^subsolutions. For holomorphic functions in Cn the first versions of the Fatou theorem are

due to E,Stein [15], E.Chirka [2], F, Forstneric [5], Y.Khurumov [8] and A.Sadullaev [14], Our approach is inspired by [17].

We will deal with some standard classes of real submanifolds of an almost complex manifold. A submanifold E of an almost complex n-dimensional (M, J) is called totally real if at every point p E E the tangent space TPE does not contain non-trivial complex vectors that is TPE n JTpE = {0}. This is well-known that the (real) dimension of a totally real submanifold of M is not bigger than n; we will consider in this paper only n-dimensional totally real submanifolds that is the case of maximal dimension. A real submanifold N of (M, J) is called generic if the complex span of TPN is equal to the whole TPM for each point p E N. A real n-dimensional submanifold of (M, J) is generic if and only if it is totally real.

Our second main result here is the following theorem.

E Q

domain Q in an almost complex manifold (M, J) of complex dimension n. Suppose that a complex function F E Lr(Q) is a djF-suhsolution on Q. Then F has an admissible limit at

E

Note that the Hausdorif n-meausure on E here is defined with respect to any metric on M;

E

result also is obtained in [17] under considerably stronger assumptions discussed above: the Q

there by the stronger condition of boundedness of djF(z) on Q.

Theorem 3.2 is established for boundaries and manifolds of class Cr though this aregularitv assumption may be highly weakend. Here we consider the question of presice regulaitv in the important special case of the standard complex structure Jst on Cn.

Theorem 3.3. Let Q be a bounded pseudoconvex domain in (Cra, Jst) with boundary bQ of class C\ Assume that Q admits a defining function which is of class C1 on a neighborhood of Q and is plurisubharmonic in Q. Let E C bQ be a generic submanifold of class C1. Suppose that a complex function F E Lr(Q) is a d jst-subsolution on Q. Then F has an admissible limit

E

E

4. Proof of Theorem 3.1

Assume that we are in the setting of Theorem 3.1. First we need the following lemma.

F F

a p-admissible curve 71 at p E E, then F has the same limit along each admissible curve in Q tangent to j1 at p.

2 1 2 line at p. Without loss of generality asssume that p = 0 (in local coordinates). Denote by p

Q

It follows by the Nijenhuis-Woolf theorem that there exists a family zt(() : D M Cn, of embedded J-holomorphic discs near the origin in Cn satisfying the following properties:

(i) the family zt is smooth on D x [0,1];

(ii) for every t E [0,1] the disc zt transversally intersects each curve 7j at a unique point coresponding to some parameter value (j(t) E D , j = 1, 2. In other words 7j(t) = zt((j(t)). Furthermore, (1(t) = 0, i.e. this point is the center of the disc zt.

In the ease of the standard complex structure each such disc is simply an open piece (suitably parametrized) of a complex line intersecting transversally the both of curves 7j. Recall that the curves are embedded near the origin and tangent at the origin so such a family of complex

J

deformation described in the proof of the Nijenhuis-Woolf theorem in Section 2, Note that for t = 1 the disc z1 intersects transversally the both curves 7j at the same point 7j (1) = p.

Furthermore, because of the condition (i), the compositions F o zt have (- derivatives of class Lp on their domains of definitions, for each p > 2 close enough to 2. Moreover, their Lp norms are bounded on D uniformly with respect to t. Indeed, it follows by the Chain Rule and (2,3) that

( F o z)s = (F-z + FzA)zl

and now we use the assumption that dJF(z) has the growth of order dist(z, bQ)-1/2+T, r > 0, j

|C2(t)l = 0(1 — t) (4.1)

as t ^ 1. The curve 71 is admissible, so we have

dist(71 (t), bQ) = 0(1 — t)

as t ^ 1. Hence, there exists p(t) = 0(1 — i) as t ^ 1 such that zt(p(t)D) is contained in Q. Applying (2.12) to the composition f := F o zt(() on the disc p(t)D, we obtain (fixing r > 0)

If(0) — f((2(t))l < (C/0(1 — t)1-2/p)(\\ f ||r + 0(1 — t) II f-c Wv)0((1 — t)1-2/p ^ 0 (4.2)

as t ^ 1. ^^^e that by (4.1) for every t the point (2(t) is ^^^teined in (1/2)p(t)D; hence, the constant C is independent of t (see the remark after (2.12)). This completes the proof. □

We continue proving the theorem. First we consider a special case where our almost complex manifold M coincides with Cn and the almost complex structure J coincides with Jst.

It suffices to consider the case where p = 0. Furthermore, after a linear change of coordinates

Q

p(z)= yn + o(IzI) (4.3)

In particular, the holomorphie tangent space H0(bQ) has the form

Ho(bQ) = {Zn = 0} (4.4)

Without loss of generality we employ the usual Euclidean distance. We have T0(bQ0) = {yn = 0}. Note that

dist(z, bQo) - Ip(z)I ^ Iy2I = dist(z,To(bQo)).

Hence we can assume 50(z) = I p(z)|. Since dist(z, H0(bQ0)) = |znI, for each a > 0 the admissible regions Aa,£(0) from (3.2) are defined by the conditions

I ZnI < (1 + a)Ip(z)I (4.5)

and

H2 < aIp(z)I1+£ (4.6)

After an additional linear change of coordinates (which preserves the previous setting) one can assume that the tangent line T0(7) is contained in the coordinate complex line Ln = (0, . . . , 0, n) n E C F

in Ln which is tangent to T0(7).

The intersection of the complex normal plane Ln with Q is the plane domain

n := {z : Z1 = ... = zn-1 = 0,yn + o(yn) < 0}

and the first inequality (4,5) defines a non-tangential region there (which tends to this half-plane

a

Fix a point (0,..., 0, z®) which satisfies (4.5). Fix a un it vector v E H°(bQ) of the form v = (v ]_,..., vn-1, 0) Consider a complex line through the point (0,..., 0, z®n) in the direction

f(v, z0n): C (C ^ 4) (4.7)

which is parallel to H°(bQ°). A simple calculation shows that the second assumption (4.6) is equivalent to the fact that f(v, z®)(rD) C Aa,£ (0) when

r~| P°n r+£ (4.8)

Clearly, this family of complex discs fill the region Aa,£(0) when (0,..., 0, z®) satisfies the first condition (4.5). Furtermore, since p° := | p(0,..., 0, z® )| ~ | y°n |, the disc f(v, z^)(p° D) is Q

The restriction F o Ln is a bounded function on n and (F o Ln)^ is of class Lp with p > 2 close enogh to 2. Furthermore, F o L^ ^ A a limit L along some rav in n with vettex at 0. By (b) Lemma 2.2 the function F o Ln admits the limit L along any non-tangential region in n. Let now z E Aa,£(0) Then there exists a unit vector v E H°(bQ) and a point zi°° in the non-tangential region on n such that the disc f(v, z®) contains the point z that is z = f(v, z®)(() for some ( with |(| ^ C|y° |1/^. Since also f(v, z°)(0) = z°, by (2.12) we have the estimate:

|F (z) — F (0,..., 0,, z°n )| = |(F o f(v, z°n))(() — (F o f(v, z°n ))(0)| ^ C | y°n ^

with t = e(1 — 2/p) > 0 Note that we apply (2.12) on a disc p°D and use (4.8) because E Aa,£(0). Since F(0,..., 0, zn) M L as y°n M 0 we conclude that F(z) M L. The case of a general almost complex structure J follows by the same argument using a slight deformation transforming the above Jsi-holomorphic discs to J-holomorphie discs. Such a deformation is always possible by the Nijenhuis-Woolf theorem and is continuous in any Ck norm. Thus, it changes only constants in estimates and the above argument literally goes through. This proves Theorem 3.1.

Now the proof of Theorem 3.2 follows exactly as in [17] using Theorem 3.1. We attach the

E E

F

proves Theorem.

In the next section we describe this construction of complex discs with the minimal boundary regularity for the case of Jst in Cn.

5. Gluing complex discs to C1 totally real manifolds

For the convenience of readers we recall here the main steps of the construction of gluing holomorphie discs to a totally real manifold of class C1. The details are contained in [18].

Everywhere we are in Cn with the standard complex structure. As usual, by a wedge-tvpe domain we mean a domain

W = {z E Cn : fa(z) < 0,j = 1,...,n} (5.1)

with the edge (or the corner)

E = {z E Cra (z) = 0,j = 1,...,n} (5.2)

We assume that the defining functions are of class Cl. Furthermore, as usual we suppose that E is a generic manifold that is d<^1 A ... A d$n = 0 in a neighborhood of E. Given 5 > 0 (which is supposed to be small enough) we also define a shrinked wedge

Wj = {ze Cn — < 0,j = 1,...,n} CW (5.3)

i=j

E C > 0

z eWs one has

C-1 dist(z, bW) ^ dist(z, E) ^ C dlst(z, bW) (5.4)

C C1 C2

from line to line.

Consider a wedge-tvpe domain (5.1) with the edge (5.2). A complex (or analytic, or holo-morphie) disc is a holomorphie map h : D ^ Cn which is at least eontinous on the closed disc D, Denote by 6D+ the upper semi-circle. We say that such a disc is glued (or attached) to a subset K of Cn along an (open, nonempty) arc 7 C 6D, if ¡(7) c K. Usually 7 wil be 6D+. Let E be an n-dimensional totally real manifold of class C1 in a neighborho od of 0 in Cn; 0e E

Q E

(vector) equation

y = h(x) (5.5)

where a vector function h = (h]_,..., hn) of class C1 in a neighborho od of 0 in Rn and satisfies the conditions

h j(0) = 0, Vh j(0) = 0, j = 1,... ,n. (5.6)

Here and below V denotes the gradient.

Fix a positive non-integer s. Consider the Hilbert transform T : u ^ Tu, associating to a real function u e Cs(6D) its harmonic conjugate function vanishing at the origin. In orther words, u + iTu is a trace on bD of a function, holomorphie on D and of class Cs(D), and satisfying T u(0) = 0

Recall that the Hilbert transform is given explicitly

Tu(eie) = 1 v.p. J^ u(eu) cot ^dt

C ( D)

any non-integer s > 0, Furthermore, for p > 1 the operator T : Lp(6D) ^ Lp(6D) is a bounded linear operator as well; we denote by \\ T \\p its norm.

Let 6D+ = {e%e : 9 e [0,^]} and bD- = {e%d : 9 e]^, 2n\} denote the upper and the lower semicircles respectively. Fix a C^-smooth real functions ^j on bD such that fy 16D+ = 0 and ^j I bD- < 0, j = 1,..., n (one may take the same function independently of j). Set ^ = (ip1,..., ^n). Consider the generalized Bishop equation

u(() = —Th(u(0) — tTp(() + c, C e bD, (5.7)

where c e Rn and t = (t 1,..., tn) e Rn tj ^ 0, are real parameters; here and below we use the notation tTip = (t 1 Tp1,..., tnTpn). The main step of our construction claims that for any p > 2, and for any c, t close enough to the origin, this singular integral equation admits a unique solution u(c, t)(() in the Sobolev class W 1,p(bD) of vector functions. Such a solution is C a( D) a = 1 — 2/

We explain also how such a solution is related to complex discs glued to E along 6D+, Indeed, consider the function

U(c, t)() = u(c, t)(() + ih(u(c, t)(0) + itip(0.

Since T2 = —Id and u is a solution of (5.7), the function U extends holomorphieally on D as a function

H(c, t)(0 = PU(c, t), (E D (5.8)

of class Ca(D), Here P denotes the Poisson operator of harmonic extension to D:

1 r 1 — |C|

2

PU(c, t)(0 = 2^J J^T—^U(c, t)(eu)dt (5.9)

The function ^vanishes on 6D+, so by (5.5) we have H (c, t)( 6D+) C E for all (c, t).

It is convenient to extend the equation (5.7) on the entire space Cn. Fix a C^ smooth function A : Rra M R+ = [0, equal to 1 on the unit ball Bra and vanishing on Rra \ 2Bra. For 5 > 0 small enough the function h(x) = A(x/5)h(x) naturally extends by 0 on the entire space Rra, Fix r > 0 small enough which will be choosen later. Then in view of (5.6) we can choose 5 = 5(t) > 0 such that the gradient Vh (x) is small on the whole Rra:

II Vh ||L~(B.n)^ T (5.10)

First we study the global equation

u(0 = —Ths(u(0) — tT^(C) + c, C E bD, (5.11)

We prove that its solutions depend continuously on parameters (c, t); this allows to localize the solutions and to conclude with the initial equation (5.7).

Let V be a domain in Rm and f E LP(V x 6D), p > 1. Then by the Fubini theorem Tf E LP(V x 6D); the variables in V are treated as parameters when the operator T acts. Hence, keeping the same notation, we obtain a bounded linear operator T : LP(V x 6D) M LP(V x 6D) with the same norm as in LP (6D), We again denote its norm by || T ||P.

Fix a domain V C R2ra of ^te parameters (c, t).

Lemma 5.1. Under the above assumptions, for any p > 1, one can choose r > 0 in (5.10), and 5 = 8(t) > 0, such that the equation (5.11) admits a unique solution u(c, t)(() E LP(Vx 6D).

The proof is contained in [18].

V

u E LPoc (R2ra x 6D), By this space we mean the space of ^functions on K x 6D for each (Lebesgue) measurable compact subset K C R2ra.

Next we study the regularity of solutions of (5.11) in the Sobolev scale.

u of class Wlfc(R2n x D)

The proof is contained in [18].

It follows by the Sobolev embedding that a solution u belongs to C 1-(2n+1')/P(V x bD), where V is an open subset in R2ra, In particular, the constructed family of discs is continuous in all

Now we note that for t = 0 the equation (5,11) admits a constant solution u(c, 0)(() = c. When c is close enough to the origin in Rn, this solution gives a point c + ih(c) E E. By continuity and uniqueness of solutions, there exists a neighborhood V of the origin in R2ra, such that for (c, t) E V any solution of (5,11) is a solution of (5,7), We obtain the following lemma.

Lemma 5.3. Given p > 2 the exists a neighborhood V of the origin in R2ra such that the Bishop equation (5.7) admits a unique solution u(c, t)(() E W 1iP(V x

Ca(V x 6D) with a = 1 — (2n + 1)/p- Note that here V depends on p (and hence, on a). Nevertheless, it follows from [3] that for each (c, t) fixed, the map ( M u(c, t)(() is of class C a( D) a < 1

Until now we did not study any geometric properties of family (5,8), Here we consider some of them which will be useful for our applications. We represent family (5,8) as a small perturbation in the W1,p norm of some model family. The model case arises when E = Rn that h=0

u(0 = —tTp(0 + C, (e bD, (5.12)

where as usual c e Rn and t=(t 1,..., tn), tj ^ 0, are real parameters. In this case the family (5.8) becomes

H(c, t)(0 = PU(c, t), (e D, (5.13)

where

U(c, t)() = —tTp(C) + c + itp(C). (5.14)

Geometrically this family of discs arises from the family of complex lines intersecting Rn along real lines; the discs are simply obtained by a biholomorphic reparametrization of the corresponding half-lines by the unit disc. These lines are given by l(c, t) : ( M t( + c, ( e C, The eonformal map —Tp + ip takes the unit disc into a smoothly bounded domain in the lower half-plane, gluing 6D+ to the real axes. One can view the parameter i as a directing vector

Their geometric properties are very simple; their detailled description (in a more general case) is contained, for example, in [17].

Let E be a totally real manifold given by (5,5), (5,6), Given d e I \ {0}, where I 3 0 is an open interval in R small enough, consider the manifolds Ed given by

y = d-1h(dx) (5,15)

Note that for every d = 0 the manifold Ed is biholomorphic to E via the isotropic dilation z M d-1z.

Set h(x,d) = d-1h(dx) when d = 0 and h(x, 0) = 0. In the last case, when d = 0, we have E0 = {y = 0} = Rn = T0(E) that is, the flat ease. Note that the function h(x, d) and its first

x d e

1 Ed

y = h(x, d), (5.16)

hj(0, d) = 0, Vxhj(0,d) = 0, de I, j = 1,...,n; (5.17)

we consider the gradient Vx with respect to x. Hence for each (c,t,d) we have the discs H( , , )

H(c,t, 0)(() coincides with the family ( 5.14). Lemma 5.4. For any p > 1 one has

\\ H (c,t, d)(0 — H (c,t, 0)(0 MV XD)M 0

d M 0

The proof is contained in [18].

The following proposition is the key technical step.

Proposition 5.1. Assume that Q C Cn is a pseudoconvex domain with C1-boundary. Let also W C Q be a wedge (5.1) with the edge E C r of type (5.2). Then one can find a family of E

(i) each disc is glued to E along bD+;

(ii) the discs fill a -shrinked wedge Ws for each, 5 > 0;

Q

Proof. Everything is local; we may assume that 0 E E and T°E = R, as in previous sections. Consider the family of discs constructed in the former section and attached to E along 6D+, The flat discs fill a prescribed wedge of type (5,1) with the edge E° = Rra, More precisely, we can fix an open convex cone K in W° = {(x, y) E R2ra : yj < 0,j = 1,...,n} with the vertex at the origin and such that K n rBn is contained in W° U {0}, for some r > 0 small enough. Clearly, the fiat discs fill a neighborhood of K n rBn. The same remains true for the cone Kz obtained by the parallel translation of K to the vertex at z E Rn. Since the family H(c, t, d)(() is a small perturbation of the fiat discs in Cs(V x D) (with any 0 < s < 1), we conclude by continuity that for d small enough the family H(c,t,d)(() also fills a presribed edge of type (5,3) with the edge Ed- By the holomorphic equivalence, the same is true for the initial edge E and a shrinked wedge Ws with anv 5 > 0, Note that in this construction we consider the discs H( , , )( )

the constant ones H(c, 0,d)(() = c. If p is a C^defining function of bQ, we use it also as a E

negative on 6D-and vanishes on 6D+. By Kerzman-Rosay [6] the domain Q admits a bounded

Q

each disc is contained in Q. □

6. Proof of Theorem 3.3

For the proof of Theorem 3,3 we need some additional geometric properties of complex discs. Each disc h : D M Q is of class Ca(D) with any a < 1. Hence, its derivatives satisfy the estimate

|dhR| ^ C(1 — |(|)

By the classical Fatou theorem we conclude that each derivative dh/d( admits a normal limit almost everywhere on bD. Hence, the image of such a normal rav is a curve j : [0, 1[m Q of class C in [0,1^, Let us prove that such a curve is admissible. By assumption of theorem, Q admits a plurisubharmonie defining function p of class C1. Note that

C-1 dlst(z, bQ) ^ p(z) ^ C dlst(z, bQ)

for all z E Q. Applying the Hopf lemma to the function p o h which is subharmonic on D, we obtain that

|(p o h)(C)| ^C(1 — |C|) (6.1)

This shows that the approach of the curve 7 to bQ is non-tangential, i.e. 7 is an admissible

F

F

limit at the point 7(1) E E.

h

h° h ( , ) vector t E R+, It defines a normal direction for each disc. Then we vary the parameter c in a neighborhood V of 0 in Rn-1 such that the boundaries of discs fill a neighborhood of the origin on E = zRn. Then the evaluation map : (c,t, () M h° (c, t)(() is a smooth diifeomorphism between V x 6D+ and a neighborhood of the origin in E. Similarly we define the evaluation map $ : (c,t, () M h(c, t)(() using the discs h attached to E. Then the map $ is a small

deformation of $o in the Sobolev W 1,p-norm; the map $ is a homeomorphism between V x 6D+ and a neighborhood of the origin in E. It follows by the well-known results (see [13]) that $ satisfies the ^-property of Lusin, i.e. the image of a set of n-measure 0 has n-measure 0. Therefore, F has admissible limits almost everywhere on E. This completes the proof.

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Alexandre Sukhov,

Institute of Mathematics,

Ufa Federal Research Center, RAS,

Chernyshevkv str. 112,

450008, Ufa, Russia

Université de Lille, Laboratoire Paul Painlevé,

Département de Mathématique, 59655 Villeneuve d'Ascq, Cedex, France, The author is partially suported by Labex CEMPI. E-mail: sukhovSmath .univ-lillel .fr