Siberian Mathematical Journal, Vol. 42, No. 5, pp. 952–960, 2001 c 2001 Myslivets S. G. Original Russian Text Copyright

ONE BOUNDARY VERSION OF MORERA’S THEOREM S. G. Myslivets

UDC 517.55

Let D be a bounded domain in Cn (n > 1) with smooth connected boundary ∂D (of class C 1 ). Suppose that the integrals of a function f ∈ C (∂D) over ∂D ∩ l vanish for all complex curves l in some class. Our aim is to answer the following question: whether f extends holomorphically to D as a function of n complex variables (i.e., whether such boundary Morera-type theorem is valid). For complex lines this question was studied by J. Globevnik and E. L. Stout in [1], M. L. Agranovski˘ı and A. M. Sem¨enov in [2], and A. M. Kytmanov and the author in [3]. In [4] this question was discussed for complex curves in C2 . Suppose that ψ = (ψ1 , . . . , ψn ) is a mapping whose components are holomorphic functions ψj in some neighborhood of the compact set KD = {w : w = ζ − z, z, ζ ∈ D} and which has a unique zero of multiplicity µ at the origin. Consider the differential form n (n − 1)! X (−1)k−1 wk dw[k] ∧ dw U (w) = , |w|2n (2πi)n k=1

where dw = dw1 ∧ · · · ∧ dwn and dw[k] is obtained from dw by deleting dwk . The differential form U (w) is the Bochner–Martinelli kernel at the point 0. By [5] the following representation is valid for every function F ∈ C 1 (D):  Z Z µF (z), z ∈ D, (1) F (ζ)U (ψ(ζ − z)) − ∂F (ζ) ∧ U (ψ(ζ − z)) = 0, z∈ / D, Dζ

∂Dζ

where the integrals over D converge absolutely and where z in U (ψ(ζ − z)) is a given point (for z ∈ D the form U (ψ(ζ − z)) is defined for all ζ ∈ D due to the condition imposed on the functions ψj , whereas if z ∈ / D then for the form U (ψ(ζ − z)) to be defined for all ζ ∈ D we must only consider points z in some neighborhood of D). Proposition 1. If ∂D ∈ C d (d ≥ 1) then each function f ∈ C l (∂D), 0 ≤ l ≤ d, is the limit in the metric of a linear combination of fractions like

C l (∂D)

m n X As (ζ − z)A (ζ − z) k

k

|ψ(ζ − z)|2n−2

k=1

,

s, m = 1, 2, . . . , n,

where z ∈ ∂D and ζ is a given point beyond ∂D. Here Ask are the cofactors of the entries Jacobian matrix of ψ. Instead of (2) we can also take fractions like X  X  n n 1 s m ψp (ζ − z)Ap (ζ − z) . ψr (ζ − z)Ar (ζ − z) |ψ(ζ − z)|2n r=1

(2) ∂ψk ∂ζs

in the

(3)

p=1

Proof. Introduce the following notations. Let θ1
, . . . , θm be n-dimensional column vectors of exterior differential forms. Denote by Dν1 ,...,νm θ1 , . . . , θm the determinant of order n whose first ν1 columns are the vectors θ1 , the next ν2 columns are the vectors θ2 , etc., and the last νm columns are the vectors θm ; ν1 + · · · + νm = n. For the properties of such determinants, we refer, for instance, to [6, Chapter 1]. The research was financially supported by the Russian Foundation for Basic Research (Grant 99–01–00790). Krasnoyarsk. Translated from Sibirski˘ı Matematicheski˘ı Zhurnal, Vol. 42, No. 5, pp. 1136–1146, September–October, 2001. Original article submitted February 17, 1997.

952

c 2001 Plenum Publishing Corporation 0037-4466/01/4205–0952 $25.00

Lemma 1. The kernel U (ψ(ζ − z)) is representable as   n X 1 ∂ 1 s U (ψ) = D1,n−1 (A , ∂ζ ψ) ∧ dζ, (n − 1)(2πi)n ∂zs |ψ|2n−2 s=1

where As is the column of the cofactors Ask , k = 1, 2, . . . , n. Proof. By [5]
1 1 U (ψ) = D ψ, ∂ ψ ∧ dψ. 1,n−1 ζ (2πi)n |ψ|2n Since

n X ∂As

k

s=1

∂zs

= 0,

(4)

k = 1, . . . , n,

we obtain   n X
∂ 1 s D1,n−1 A , ∂ζ ψ ∧ dζ ∂zs |ψ|2n s=1   n X
∂ 1 = D1,n−1 As , ∂ζ ψ ∧ dζ 2n−2 ∂zs |ψ| s=1

= −(n − 1)

n X

n P

k ψ k ∂ψ n ∂zs X k=1

s=1

|ψ|2n

(−1)j−1 Asj Djn−1 (∂ζ ψ) ∧ dζ

j=1

n X ψs = (n − 1) (−1)s−1 Dsn−1 (∂ζ ψ) ∧ dψ |ψ|2n s=1


(n − 1) = D1,n−1 ψ, ∂ζ ψ ∧ dψ = (n − 1)(2πi)n U (ψ). 2n |ψ| Here Djn−1 (∂ζ ψ) is the determinant of order (n − 1) that results from deleting the first column and the
jth row in the determinant D1,n−1 As , ∂ζ ψ .  Lemma 2. The kernel U (ψ) is representable as n n (n − 2)! X ∂ X (−1)m−1 U (ψ) = (2πi)n ∂zs |ψ|2n−2 s=1

m=1

n X

! m Ask Ak

¯ dζ[m] ∧ dζ.

k=1

Proof. The assertion follows from Lemma 1 and the identity s

D1,n−1 (A , ∂ζ ψ) = (n − 1)!

n X m=1

m−1

(−1)

n X

! m Ask Ak

¯ dζ[m]. 

k=1

We can easily derive Proposition 1 from (1) and Lemmas 1 and 2. Indeed, consider a sufficiently small neighborhood V of ∂D (such that all functions ψj (ζ − z) are well defined). Extend f to a function on V of the class C l with compact support in V . Approximating f in the C l metric in V by functions of the class C ∞ , we may assume that f itself is infinitely differentiable. Applying (1) to f in V , we obtain Z − ∂f (ζ) ∧ U (ψ(ζ − z)) = µf (z), z ∈ ∂D. Vζ

953

Executing the change of variables ζ = z + w, we obtain Z − ∂f (z + w) ∧ U (ψ(w)) = µf (z),

z ∈ ∂D.

Cn

We can find the derivatives with respect to z and z¯ up to the order l by differentiating this equality under the integral sign (due to absolute convergence of the integral). Choosing a sufficiently small neighborhood V 0 of ∂D, we can therefore make the integral over V 0 arbitrarily small in the C l metric. In the integral over V \ V 0 we replace the integrand with an integral sum and replace the derivatives with difference quotients (applying Lemmas 1 and 2). The so-obtained fractions are arbitrarily close to f in the C l metric. Denseness of fractions (3) is immediate from representation (2) on using fractions (2).  Corollary 1. If f ∈ C (∂D) satisfies the moment conditions ! Z m n X Ask (ζ − z) Ak (ζ − z) ¯ p] ∧ dζ = 0 f (ζ)∂ ζ ∧ dζ[j, |ψ(ζ − z)|2n−2

(5)

k=1

∂D

for all z ∈ / ∂D and j, s, m, p = 1, . . . , n then f extends to a function F ∈ C (D) holomorphic on D. Proof. From Proposition 1 we conclude that Z ¯ p] ∧ dζ) = 0 f (ζ)∂ ζ (α(ζ) ∧ dζ[j, ∂D

for every smooth function α(ζ) in a neighborhood of ∂D. Consequently, f is a CR-function on ∂D. Since ∂D is connected, f extends holomorphically to D.  Corollary 1 is one of the versions of the Hartogs–Bochner theorem. Note that, in this assertion, it suffices to require validity of (5) only at points z in some open set V ⊂ D or V ⊂ Cn \ D. As we see below, the condition of Morera’s theorem transforms into the following orthogonality condition: Z f (ζ)Φ(ζ − z)U (ψ(ζ − z)) = 0, z ∈ / ∂D, (6) ∂Dζ

for f ∈ C (∂D) and a function Φ like Φ(w) = ϕ1 (w)ψ1 (w) + · · · + ϕn (w)ψn (w),

(7)

where the functions ϕj (w) (as well as ψj (w)) are holomorphic in some neighborhood of the compact set KD or ϕj are meromorphic functions such that the form ΦU (ψ) has no singularities for ζ 6= z. We first study (6) for a special choice of Φ in (7). Lemma 3. We can rewrite (6) as   Z 1 f (ζ)∂ ζ D1,1,n−2 (ϕ(ζ − z), ψ(ζ − z), ∂ζ ψ) ∧ dψ = 0, |ψ(ζ − z)|2n−2 ∂Dζ

where z ∈ / ∂D and ϕ is the column of the functions ϕj , j = 1, . . . , n. ϕ Proof. Consider the columns τ = Φ and   ψn ψ1 ψ . ,..., η= = |ψ|2 |ψ|2 |ψ|2 954

By Lemma 1 of [5] (or directly), U (ψ(ζ − z)) =

1 ∂ ζ D1,1,n−2 (τ (ζ − z), η(ζ − z), ∂ ζ η(ζ − z)) ∧ dψ (2πi)n

beyond the zeros of the function Φ(ζ − z). Using the homogeneity property of the determinant D of differential forms, we obtain

=

1 ∂ζ (2πi)n



Φ(ζ − z)U (ψ(ζ − z))   1 D1,1,n−2 ϕ(ζ − z), ψ(ζ − z), ∂ζ ψ(ζ − z) ∧ dψ.  |ψ(ζ − z)|2n−2

Lemma 4. We can rewrite (6) as     n Z X
∂ 1 s f (ζ)∂ ζ D1,1,n−2 ϕ, A , ∂ζ ψ ∧ dζ = 0 ∂zs |ψ|2n−4

(8)

s=1∂D

if z ∈ / ∂D and n > 2; and 2 Z X

 f (ζ)∂ ζ

s=1∂D


∂ log |ψ|2 D1,1 (ϕ, As ) ∧ dζ = 0 ∂zs

(9)

if n = 2. Proof. The proof is the same as that of Lemma 1. In this lemma we only need equality (8), since (9) was studied in [4]. Thus, we assume that n > 2. First of all, we show that, for a special choice of the functions ϕj , we can factor out the derivatives with respect to zs from the integral sign in (8). Let the column vector ϕ look like ϕ(w) = J1 Ak , where J is the determinant of the Jacobian matrix of ψ (with J 6≡ 0) and Ak is the column of the cofactors Akm , m = 1, . . . , n, k = 1, . . . , n. Lemma 5. We can rewrite (6) as   k   X ∂ Z 1 A s f (ζ)∂ ζ D1,1,n−2 , A , ∂ζ ψ ∧ dζ = 0 ∂zs |ψ|2n−4 J s6=k

(10)

∂D

if z ∈ / ∂D. Although the determinant J may vanish on some surface, the determinants D under the integral sign in (10) have no singularities as we will see from the proof of Lemma 5. Proof. Put ϕ = J1 A1 . To prove the lemma, it suffices (in view of (8)) to demonstrate that  1  n X ∂ A s D1,1,n−2 , A , ∂ψ = 0. ∂zs J s=2

By Laplace’s theorem,  1  X n n X ∂ A ∂ X (−1)p+r s D1,1,n−2 , A , ∂ψ = ∂zs J ∂zs p
s=2

1 Ap A1 r

Asp p,r D (∂ψ), Asr n−2

where Dp,r n−2 is the determinant that results from deleting the first two columns and the rows with numbers p and r in D1,1,n−2 . 955

Using the properties of determinants of cofactors (see [7, p. 31]), we obtain 1 Ap Asp 1,s A1 As = J · Ap,r , r r

(11)

where A1,s p,r are the cofactors of the minor at the intersection of the first and sth columns and the pth and rth rows in the Jacobian matrix of ψ. On the other hand, it is easy to show that n X ∂ 1,s A = 0. ∂zs p,r

(12)

s=2

From here and (11) we obtain the required equality. Moreover, (11) shows that the determinants D1,1,n−2 in (10) have no singularities.  It follows from the proof of Lemma 5 that (10) amounts to the following: Z n X ∂ f (ζ)∂ ζ βk,s ∧ dζ = 0 ∂zs s6=k

where βk,s =

for z ∈ / ∂D,

(13)

∂D

XX 1 l,m ¯ (−1)p+r Ak,s p,r (ζ − z)Ap,r (ζ − z)dζ[l, m]. |ψ(ζ − z)|2n−4 p
Therefore, we have to find out whether linear combinations of fractions more general than in Proposition 1 are dense in the class C k (∂D). Proposition 2. Suppose that n > 2 and ∂D ∈ C d . The linear combinations of fractions like Ql,s,m,k (ζ − z) =

X 1≤p
m,k Al,s p,r (ζ − z)Ap,r (ζ − z) , |ψ(ζ − z)|2n−4

z∈ / ∂D, ζ ∈ ∂D, k, s, m, l, = 1, . . . , n, are dense in the space C u (∂D), 0 ≤ u ≤ d. Proof. From (12) we deduce X ∂ X ∂ Al,s As,l = 0 p,r − ∂zl ∂zl p,r l
(14)

l>s

for all s, p, r = 1, . . . , n. Indeed, if interchange z1 and zs in (12) and the so-obtained expression to their places then we arrive at (14). Assume that p < r. Then  s  Ar if X ∂ψq X ∂ψq Al,s − As,l = −Asp if ∂zl p,r ∂zl p,r  l>s l
afterwards put the first columns of

q = p, q = r, q 6= p, r.

(15)

The above identity ensues from the ordinary rule for expansion of determinants in a row with the signs s of the cofactors Al,s p,r and Ap taken into account. Using (14) and (15), for given s, m, and k we obtain X ∂ X ∂ Ql,s,m,k − Qs,l,m,k ∂zl ∂zl l
956

l>s

PP P = −(n − 2)

p
l
P = −(n − 2)

ψq

p
∂ψq l,s ∂zl Ap,r

−

P

ψq

l>s 2n−2 |ψ|

∂ψq s,l
m,k ∂zl Ap,r Ap,r


m,k ψ p Asr − ψ r Asp Ap,r |ψ|2n−2

= Rs,m,k .

Applying (14) and (15) again, for given s and k we now obtain X ∂ ∂ Rs,m,k − Rs,k,m ∂¯ zm ∂¯ zm mk ! " X ∂ψp m,k X ∂ψp k,m (n − 2) X s A − A Ar = − 2n−2 ∂zm p,r ∂zm p,r |ψ| pk mk "
X m,k X ∂ψq (n − 2)(n − 1) X s s + ψ p Ar − ψ r Ap Ap,r ψq |ψ|2n ∂zm p
m>k

+


(n − 2)(n − 1) X k k
ψ p Asr − ψ r Asp ψp Ar − ψr Ap 2n |ψ| p
Replacing derivatives with difference relations and applying Proposition 1, we finish the proof.  Theorem 1. Suppose that ∂D ∈ C 2 and f ∈ C (∂D). If Z f (ζ)Φk (ζ − z)U (ψ(ζ − z)) = 0 for all z ∈ / ∂D, k = 1, . . . , n,

(16)

∂Dζ

where

n

X 1 Φk (ζ − z) = Aks (ζ − z)ψs (ζ − z), J(ζ − z) s=1

then f extends to a function F ∈ C (D) holomorphic on D. Condition (16) (as (4) shows) amounts to the following: n P

Z f (ζ)

s=1

Aks (ζ − z)ψs (ζ − z) |ψ(ζ − z|2n

D1,n−1 (ψ, ∂ζ ψ) ∧ dζ = 0;

∂Dζ

therefore, the integrand in (16) has no singularities for ζ 6= z despite the presence of the Jacobian J in the denominator. Proof. The case of two variables was settled in [4]. Assume that n > 2. We write down the conditions of the theorem like (13). Using Lemma 5 and Proposition 2, we approximate the function 957

|ζ − z|4−2n , ζ ∈ ∂D and z is given beyond ∂D, in the class C 2 (∂D) by linear combinations of the fractions of Proposition 2. Then from (16) we obtain   X ∂ Z
1 k es e f (ζ)∂ ζ D1,1,n−2 A , A , ∂ζ (ζ − z) ∧ dζ = 0, ∂zs |ζ − z|2n−4 s6=k

∂D

˜ − z) = ζ − z. Lemmas 3 e k are the corresponding cofactors for the identity mapping ψ(ζ z∈ / ∂D, where A and 4 show that we can rewrite this condition as Z f (ζ)(ζk − zk )U (ζ − z) = 0, z ∈ / ∂D, k = 1, . . . , n, ∂D

where U (ζ − z) is the Bochner–Martinelli kernel. Applying the Laplace operator ∆=

∂2 ∂2 + ··· + ∂z1 ∂¯ z1 ∂zn ∂¯ zn

to the left-hand side of the last equality and using harmonicity of the coefficients of the Bochner–Martinelli kernel, we obtain Z ∂ f (ζ)U (ζ − z) = 0, z ∈ / ∂D, k = 1, . . . , n. ∂¯ zk ∂D

Thus, the Bochner–Martinelli integral of f is a holomorphic function outside the boundary of D. Since ∂D is connected and the integral tends to 0 as |z| → ∞, this integral vanishes outside D. By the jump theorem for the Bochner–Martinelli integral (see, for instance, [8, Chapter 1]), we conclude that the sought holomorphic extension is given by the Bochner–Martinelli integral.  We now turn to obtaining an analog of Morera’s theorem. We consider the class of complex curves Lz,b like Lz,b = {ζ ∈ Cn : ζ1 = z1 + tk1 , ζj = zj + bj tkj χj (tk1 ), j = 2, . . . , n, t ∈ C}, where k1 , . . . , kn ∈ N, b = (1, b2 , . . . , bn ), and χj (τ ), j = 2, . . . , n, are holomorphic functions in a neighborhood of the projection of the compact set KD to the coordinate plane z1 ; moreover, these functions vanish nowhere. If we fix z in some neighborhood of the closure of D then, for every point ζ in the same neighborhood such that z1 6= ζ1 , there is a curve Lz,b passing through ζ (for a suitable choice of b). All curves Lz,b for a given z intersect at 0. If they intersect at another point then we can easily demonstrate that the jth coordinates of the vectors b for them are obtained from each other by rotation through an angle which is a multiple of k11 2πkj . Therefore, to determine b uniquely, we assume that the argument of bj satisfies the condition 0 ≤ arg bj < 2πrj , j = 2, . . . , n, (17) where rj is the fractional part of the number k11 kj (if kj is divisible by k1 then we impose no conditions on arg bj ). Essentially, Lz,b is a parameterization of the following complex curves defined explicitly: {ζ : ζj = kj

zj + bj (ζ1 − z1 ) k1 χj (ζ1 − z1 ), j = 2, . . . , n}. Consequently, for a given z we obtain the fibering of the neighborhood D \ {ζ : ζ1 = z1 } into the curves Lz,b for vectors b satisfying (17). Now, Sard’s theorem shows that, for almost every b satisfying this condition, the intersection of Lz,b with ∂D either is empty or is the union of finitely many closed piecewise smooth curves. We introduce the holomorphic functions p

ψ1 (ζ) = ζ1p1 , 958

ψj =

ζj j p

χj j (ζ1 )

,

j = 2, . . . , n,

where the natural numbers pj are chosen so that p1 k1 = · · · = pn kn = p. These functions are holomorphic in a neighborhood of the compact set KD and have the only common zero of multiplicity µ = p1 . . . pn , the origin. Therefore, Theorem 1 is valid for these functions. We consider the kernel U (ψ(ζ − z)) in the new coordinates t and b. It looks like dt U (ψ ∗ (t, b)) = ∧ λ(b), t where p(n − 1)! (−1)n−1 d¯bp22 ∧ · · · ∧ d¯bpnn ∧ dbp22 ∧ · · · ∧ dbpnn  n λ(b) = . n P (2πi)n 2p 1+ |bj | j j=2

ψ ∗ (t, b)

Here is the composite of the mapping ψ(ζ − z) and the mapping ζ − z determining the curves Lz,b . Here and in the sequel, the symbol ∗ stands for the passage from the variables ζ to the new variables (t, b) in the definition of the curves Lz,b . p Indeed, ψ1∗ (t, b) = tp and ψj∗ (t, b) = bj j tp , j = 2, . . . , n. Therefore, ! n X 2 ∗ 2 2p 2pj |ψ(ζ − z)| = |ψ (t, b)| = |t| 1+ |bj | . j=2

We have

dψ ∗ = dψ1∗ ∧ · · · ∧ dψn∗ = ptpn−1 dt ∧ dbp22 ∧ · · · ∧ dbpnn

and n X

k−1

(−1)

ψk∗ dψ ∗ [k]

=t

p

dψ2∗

∧ ··· ∧

dψn∗

+

k=1

n X

(−1)k−1 bpkk tp dψ ∗ [k]

k=2

= tpn dbp22 ∧ · · · ∧ dbpnn + ptpn−1

n X

p

(−1)j−2 bj j dt ∧ dbp22 ∧ . . . [j] · · · ∧ dbpnn

j=2

+

n X

! (−1)k−1 bpkk tpn−1 dt ∧ dbp22 ∧ . . . [k] · · · ∧ dbpnn

= tpn dbp22 ∧ · · · ∧ dbpnn .

k=2

We now find out how condition (16) of Theorem 1 looks for the mapping ψ, in particular, we find the functions Φj in the new coordinates t and b. Denote the functions 1pj by γj , j = 2, . . . , n. We have χj

p1 −1

pn −1

J(ζ − z) = p1 . . . pn (ζ1 − z1 ) . . . (ζn − zn ) γ2 (ζ1 − z1 ) . . . γn (ζ1 − z1 ). s The column vectors A are as follows: A1 = (p2 . . . pn (ζ2 − z2 )p2 −1 . . . (ζn − zn )pn −1 γ2 . . . γn , 0, . . . , 0), A2 = (−p3 . . . pn (ζ2 − z2 )p2 (ζ3 − z3 )p3 −1 . . . (ζn − zn )pn −1 γ20 γ3 . . . γn , p1 p3 . . . pn (ζ1 − z1 )p1 −1 (ζ3 − z3 )p3 −1 . . . (ζn − zn )pn −1 γ1 γ3 . . . γn , 0, . . . , 0), . . . , An = (−p2 . . . pn−1 (ζ2 − z2 )p2 −1 . . . (ζn−1 − zn−1 )pn−1 −1 (ζn − zn )pn γ2 . . . γn−1 γn0 , 0, . . . , p1 . . . pn−1 (ζ1 − z1 )p1 −1 . . . (ζn−1 − zn−1 )pn−1 −1 γ1 . . . γn−1 ). Calculating the functions Φj , we obtain ζ1 − z1 ζ2 − z2 (ζ1 − z1 )(ζ2 − z2 )γ20 Φ1 (ζ − z) = , Φ2 (ζ − z) = − ,..., p1 p2 γ2 p1 p2 ζn − zn (ζ1 − z1 )(ζn − zn )γn0 Φn (ζ − z) = − . pn γn p1 pn In the coordinates t and b these functions look like b2 t k2 bn t kn t Φ∗1 (t, b) = (tk1 )0 , Φ∗2 (t, b) = (t χ2 (tk1 ))0 , . . . , Φ∗n (t, b) = (t χn (tk1 ))0 . p p p 959

Theorem 2. Suppose that ∂D ∈ C 2 and a function f ∈ C (∂D) satisfies the conditions Z f ∗ (t, b) d(tkj χj ) = 0 ∂D∩Lz,b

for all j = 1, . . . , n, almost all points z in a neighborhood of D, and almost all vectors b satisfying (17). Then f extends holomorphically to D as a function F in the class C (D) (we assume that χ1 = 1 for j = 1). Proof. From Theorem 1, the shape of the form U (ψ) and the functions Φj in the coordinates t and b, and Fubini’s theorem we see that Z Z Z f (ζ)Φj (ζ − z)U (ψ(ζ − z)) = λ(b) f ∗ (t, b) d(tkj χj ) = 0. ∂Dζ

Cn−1

∂D∩Lz,b

Theorem 2 generalizes the boundary Morera theorem of [1] which treated the case of complex lines Lz,b . If χj ≡ 1, j = 1, . . . , n, then Theorem 2 transforms into the boundary Morera theorem for algebraic curves. As a corollary we derive a generalization of Theorem 3 of [3] about functions with the one-dimensional holomorphic continuation property along complex curves. A function f ∈ C (∂D) possesses the one-dimensional holomorphic continuation property along complex curves like Lz,b if, for every curve Lz,b such that Lz,b ∩ ∂D 6= ∅, there is a function Fz,b (t) such that: (a) Fz,b ∈ C (D ∩ Lz,b ); (b) Fz,b = f on the set ∂D ∩ Lz,b ; (c) Fz,b is a holomorphic function in t at the interior points of the set D ∩ Lz,b (with respect to the topology of Lz,b ). Corollary 2. If ∂D ∈ C 2 and a function f ∈ C (∂D) possesses the one-dimensional holomorphic continuation property along complex curves Lz,b then f extends holomorphically to D. This assertion generalizes the well-known Stout theorem [9] about functions with the one-dimensional holomorphic continuation property along complex lines and Theorem 3 of [3]. References 1. Globevnik J. and Stout E. L., “Boundary Morera theorems for holomorphic functions of several complex variables,” Duke Math. J., 64, No. 3, 571–615 (1991). 2. Agranovski˘ı M. L. and Sem¨enov A. M., “Boundary analogs of Hartog’s theorem,” Sibirsk. Mat. Zh., 32, No. 1, 160–170 (1991). 3. Kytmanov A. M. and Myslivets S. G., “On a certain boundary analog of Morera’s theorem,” Sibirsk. Mat. Zh., 36, No. 6, 1350–1351 (1995). 4. Kytmanov A. M. and Myslivets S. G., “On holomorphic continuation of functions along complex curves and an analog of Morera’s theorem,” in: Complex Analysis, Differential Equations, and Numerical Methods and Some of Their Applications [in Russian], Inst. Mat. (Ufa), Ufa, 1996, Vol. 2, pp. 71–77. 5. Kytmanov A. M. and Myslivets S. G., “On holomorphy of functions representable by the logarithmic residue formula,” Sibirsk. Mat. Zh., 38, No. 2, 351–361 (1997). 6. A˘ızenberg L. A. and Dautov Sh. A., Differential Forms Orthogonal to Holomorphic Functions or Forms, and Their Properties [in Russian], Nauka, Novosibirsk (1975). 7. Gantmakher F. R., The Theory of Matrices [in Russian], Nauka, Moscow (1967). 8. Kytmanov A. M., The Bochner–Martinelli Integral and Its Applications [in Russian], Nauka, Novosibirsk (1992). 9. Stout E. L., “The boundary values of holomorphic functions of several complex variables,” Duke Math. J., 44, No. 1, 105–108 (1977).

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