CC BY
5
УДК 517.55
Holomorphic Extension of Continuous Functions Along Finite Families of Complex Lines in a Ball
Alexander M. Kytmanov* Simona G. Myslivets^
Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041
Russia
Received 10.06.2015, received in revised form 01.07.2015, accepted 25.07.2015 In this paper we consider continuous functions given on the boundary of a ball B of Cn, n > 1, and having one-dimensional property of holomorphic extension along the families of complex lines, passing through finite number of points of B. We prove the existence of holomorphic extension of such functions in the ball B.
Keywords: holomorphic extension, Poisson kernel, complex lines. DOI: 10.17516/1997-1397-2015-8-3-291-302
Introduction
This paper presents some results related to the holomorphic extension of functions f, defined on the boundary of a ball B c Cn, n > 1, in this ball. We deal with a functions with the one-dimensional holomorphic extension property along the complex lines.
The first result related to our subject was received by M.L.Agranovsky and R.E.Valsky in [1], who studied the functions with a one-dimensional holomorphic continuation property on the boundary of a ball. The proof was based on the automorphism group properties of a sphere.
E.L.Stout in [2] used complex Radon transformation to generalize the Agranovsky and Valsky theorem for an arbitrary bounded domain with a smooth boundary. An alternative proof of the Stout theorem was obtained by A.M.Kytmanov in [3] by applying the Bochner-Martinelli integral. The idea of using the integral representations (Bochner-Martinelli, Cauchy-Fantappie, multidimensional logarithmic residue) has been useful in the study the functions with one-dimensional holomorphic continuation property (see review [4]).
The problem of finding the different families of complex lines, sufficient for holomorphic extension was posed in [5]. Clearly, the family of complex lines passing through one point is not enough. As shown in [6], the family of complex lines passing through a finite number of points also, generally speaking, is not sufficient.
In [6] we proved that the family of complex lines crossing the germ of generic manifold y, is sufficient for the holomorphic extension. In [7] we consider a family of complex lines passing through the germ of a complex hypersurface. In particular, C2, it can be any real analytic curve. Various other families are given in [8-11]. We note here the work [9,11], where it is shown that a family of complex lines passing through somehow located a finite number of points is sufficient for holomorphic extension. But it is approved only for real-analytic or infinitely differentiable functions defined on the boundary. Then in C2 Agranovsky and Globevnik for real-analytic
*akytmanov@sfu-kras.ru tasmyslivets@sfu-kras.ru © Siberian Federal University. All rights reserved
functions defined on the boundary shown that it is enough two points lying in the closure of the ball. The Globevnik example shows that for continuous functions on the boundary of two points is not enough for holomorphic extension.
In [12-15] it is shown that for a class of continuous functions given on the boundary of a ball the family of complex lines passing through finite points in the ball is sufficient. Baracco was the first to prove this result which earlier explicitly conjectured in [9]. Globevnik [13] gave an alternative proof, even the case when the vertices lie outside of the ball. Authors in [14,15] for proof use the Poisson kernel and the automorphisms of a ball.
Those results it is given by completely different methods.
In this paper we give a new proof of results from [14,15] without use the automorphisms of a ball.
1. Invariant Poisson kernel of a ball
Let B = {z £ Cn : |z| < 1} be the unit ball in Cn, n > 1, centered at the origin and let S = dB be a boundary of the ball. Consider the invariant Poisson kernel [17, p. 48]
( (1 -|z|2)n = (1 -<z,C>)n
P (z Z ) — cn ^ ( I 2n — Cn
|1 -<z,c )|2n (1 -(z,or (1 -<c,z->r (n - 1)!
where Cn = 0 n ', (z, Z) = ziZi +-----+ ZnZn.
2nn _
If the function f (z) is M-harmonic in B and continuous on B, then we have the integral representation
F(z)= / f (Z)P(z,Z) da(Z), (1)
Js
where
2 n
) = (-1)fc-1a dC[k] a dz (2)
in z—/
k=1
is Lebesgue measure on S, dZ = d^1 A • • • A d£n, df[k] = dC1 A • • • A d(iI-1 A dCk+1 A • • • A dCn. The boundary values of function F(z) coincides with f(Z), i.e., F(z) = f(Z). Recall that M-harmonic function satisfies the Laplace equation [17, p. 55-56]
A F (z) =0,
where
AF(z) = 4(1 - |z|2) ¿ (Sjk - zjCk) jk
j,k=1 j k
and Sjk is Kronecker symbol. The functions holomorphic in the ball B are M-harmonic. In particular, the formula (1) is an integral representation for holomorphic functions. Consider the complex line of the form
lz,b = {Z £ Cn : Z = z + bt, t £ C}, (3)
where z £ Cn, b £ CPn-1. Complex lines passing through two points a and c is denoted by 1a,c.
We will say that a function f £ C(S) has the one-dimensional holomorphic extension property along the complex line 1z,b, if S n 1z,b = 0 and there exists a function F;z t with the following properties:
1) F,zt £C(B n 1z,b),
2) Flzb = f on the set S n lzb,
3) function Flz b is holomorphic at the interior (with respect to the topology of lz,b) points of set B n lz,b.
Let r is a set in Cn. Denote by Lr the set of all complex lines lz,b such that z e r, and b e CPn-1, i.e., the set of all complex lines passing through z e r.
We will say that a function f e C(S) has the one-dimensional holomorphic extension property along the family Lr, if it has the one-dimensional holomorphic extension property along any complex line lz,b e Lr.
We will say, that set Lr is sufficient for holomorphic extension, if the function f e C(S) has the one-dimensional holomorphic extension property along all complex lines of the family Lr, and then the function f extends holomorphically to B (i.e., f is a CR-function on S). The different families of sufficient sets were considered by Agranovsky, Valsky, Stout, Globevnik, Baracco and the authors of this article.
2. Complexification of the Poisson kernel and its applications
Consider the kernel of the form
(l _ (z w)\n
Q(z ,w ,Z ) =Cn (1 _(z (C w))n ■ (4)
Obviously, that P(z, Z) = Q(z, z, Z). We introduce the function
$(z,w) = f f (Z )Q(z,w,Z) da(Z). Js
This function is holomorphic in the variables (z,w) in B x B c C2n, because the denominator in the kernel (4) does not vanish, when Z e S and z,w e B. Note that $(z, z) = F(z), and the derivatives
d a+ß $
dzadwp
where
da+ß F
(5)
w=2 dzadzß
da+ß$ d llal + lßl$
dzadwp dza ■ ■ ■ dzandwf1 ■ ■ ■ dwfc and a = (a1,..., an), [ = ([1,..., [n) are multi-indices. We denote ||a|| = a1 + ■ ■ ■ + an, ||3|| = [1 + ■■■ + 3n.
Theorem 1. Let a function f (Z) e C(S) have the one-dimensional holomorphic extension property along the family L{0}, then for the integral
$(z,w) = f f (Z)Q(z,w,Z) da(Z) Js
d a$(0, w)
true the properties $(0, w) = const and the derivatives-- are polynomials in w of degree
not higher than | a| .
Proof. Consider the derivative
da+P f 1 \_ Cap CaZP
dzadwß\ (1 - (z,Ç))n (1 - (Ç,w))nJ (1 - (^f)) n+l|a|l( 1 - (Cw)) n+llßl1
where Za = Ci1 • • • , Ca = Ci1 • • • , and Ca,g are some constants.
da+g Q(o, w)
It is easy to show that the derivative of —-——\— is the sum of terms of a form
\ ;; dzadwg
Ca g S Ca-Y' Z ^-7" W
—a'g'7'-—ii-TTj—t—— at ||Jl| < n, ||y''|| < 117'y, and a > 7', ft > 7". Therefore, the derivative
(1 - (C,w>)n+"g" "7 11
da+g$(0,w) . .
is a linear combination of integrals
dzadwe
za-7' zв-y''
f(Z)--Z Л+iieii-iiY'ii )• (6)
(1 -<C,w»
The form da(C) in variables b and t, where С = bt, t G C, was calculated in ( [4])
da(bt) = in|t|2n-2tdt Л ^¿(-1)k-1bfc db[kA Л ^¿(-1)k—1 b fc db[kA .
% 4=1 ' 4=1 '
Since on the sphere S the equality 1 = |C| = |bt| holds, then |t| = and t = • So da(bt) = in¿n dt Л (¿(-1)k-1bfc db[kA Л (¿(-1)k-1bfc = A(b) Л
1 1 4=1 ' 4=1 '
From Fubini's theorem we obtain
/С a—7' z в—y''
f(C)---Z+........... da(C) =
S (1 -(C,w))
/с ^|H|-||Y'||t||в||-||Y''||
A(bW f(bt)--хП+||вц-||7''|| dt =
J t( 1 - t(b,w))"+||в" ||y "
CP"-1 Snio.b V X ' ''
/Г t|e|-|7''!
A(bM f(bt) ......... ..ч»+||в||-||7''|| dt = 0,
, J tHHMK1 1 - t(b,w) "7 "
if УвII > ||a|| (then ||в|| - ||y''h > ||®H - ||y'||), and the function --1-- is holomorphic in the
1 - t(b, w)
closed disk |t| < —, such that B П /0,b. Hence by virtue of (6)
d а+в Ф(0, w) = 0 (7)
dzadwe (7)
at ||в| > H"!
Therefore, by the Taylor formula for the function Ф(.г, w) at the point (0,0) we get that
даФ(0, w)
Ф(0, w) = const and the derivatives —d~a- are polynomials in w of degree not higher
than ||a||. □
Corollary 1. Under the hypotheses of Theorem 1 the equality
да+вF(0, 0)
dzadze
holds if ||в|| > ||a|.
Proof. Substituting in Eq. (7) w = z and using Eq. (5), we obtain the desired result.
□
Theorem 2. Let the function f (Z) G C(S) have the one-dimensional holomorphic extension
даФ(а, w)
property along the family £{a}, a G B, a = 0, then Ф(а, w) = const and the derivatives--
are polynomials in w of degree not higher than ||a||.
Proof. Consider the function §(z,w) and a complex line 1а,ь. Since дв <^(z,w) [ дв Q(z,w,Z)
dwß j S
Г ( dßQ(z, w, Z) )
= JS f (Z) dwß da(Z 1
we calculate
dßQ(z,w,Z) _ dß .чп 1
((1 -(z,w))
dwß 3wß\y N' '7 (1 -{z,Z))n (1 - (Z,w))r
_ ßß—1 / 1 \ д^
dw^l (T-WÔ-ÎC^T J dw ^ - (z,w)n =
0<7<ß dwß—7\(1 -(z,z)T(1 -(Z,w)Y
byCß(-1)b\\Zß—~'n(n - 1) • ... • (n - ||7||)(1 - (z,w))n—MzY
O^Y^ß
^ (1 -(z,z)T (1 -(Z,w))n+^
Z ß—Y{ 1 -(Z,w))^ 7"z' ч - ¡1 4 - (Z,w
^ dßl\ 1-Шn( 1 -(Z,w))n+^ß—r
Let the multi-index ß = (ßi,..., ßn-i) such that ||ß'|| = ßi + ■ ■ ■ + ßn-i > 0. Using a unitary transform, we will transfer the point a in the point (0,..., 0, |a|). Under this transform the kind of the form da does not change (see [16, Lemma 2.7]). We get
dßQ(a, w, a + bt) (a + bt)ß-Y (1 -(a,w))n "7" aY
dWß =Cn dßY( , -,rn\n(
Jw (1 — {a, a + bt)) (1 — {a + bt, w))
at Z = a + bt and z = a. The expression (a + bt)ß Y means (ai + bit)ßl n ■ ■ ■ (an + bnt)ßl Since B n la,b is a circle
K = 11 G C :
t +
(a, b)
|b|2
<
(1 - |a|2) |b|2 + j (a, b) |
|b|4
(a, b) |b|2
boundary of this circle , provided that |a + bt| = 1:
with center at the point t0 = —nrr , then we consider the restriction of the form da on the
п
da = ^(-1)к—ЧкdZ[k] Л dZ =
k=1
п
)k—1(a,. + h,Ztn—1tn—1dh[k] Л dt Л Y^(-1)j—1b,
= J3(-1)k—1 (ak + b кt)tn—1tn—1db[k] Л dt Л Y,(-1)j—1bjdb[j].
k=i j=1
C
п
C
2
2
- 1 - t(a,b) —|a|2 , Smce t = . Z ' ' ' then (a, b) +11 b |2
1 t((1 — |a|2)|b|2 + |(a, b)|2)
1 — (a, a + bt) = --—=--——-.
(a, b) + 11 b |2
Therefore
*(a'w) = cni f (Z) V ^ (a + bt)P-Y(1 — (a>w)) Js 'tr^. h — (a, a + bt ) n 1 — (a + b
Y{ 1 — (aw)1" "7"aY
dwP ^ A / - . T,T\\n (1 / \\»+ll/S-7ll X
S (1 — (a, a + bi)J (1 — (a + bt, w)J
n n
x ( —1)k-1(afc + bfci)in-1tn-1db[k] A dt A ^( —1)j-1bjdb[j]. k=i j=1
Since t = 1 ^ t(a;b) ^ H .g performed on the boundary of the circle K, then power of de-(a, b) + t| b| 2
nominator (a, b) + 11 b |2 goes to numerator of all fraction and it reduced with denominator of
Z.
From the form of the point a we get the expression (a + bt)P Y contains a factor t in the
, , Wa,w) positive degree. By assumption of theorem --—-5— = 0 and $(a, w) = const.
dwp
Qa+Pw)
Similar it is proved that the derivatives
dzadwp
0 if HftH > ||a|| and therefore the
da$(a, w)
derivatives —- are polynomials on w of degree not higher than ||a||.
3. Decomposition of the Poisson kernel
We represent the function $(z, w) as a sum of homogeneous polynomials in z and w. We decompose Q(z, w, Z) by powers of (z, Z), (Z,w). Since
1 o
C"+fc-1(z, Z)k,
(1 — (z,zV))n ¿0 n+k-1
.. oo
(1 1 ))n = £cn+1-1(Z,w)k
(1 — (Cw)) ^
(considered the series converge absolutely for Z G S, z,w G B, and uniformly on S x K, where K is arbitrary compact set in B x B), then
oo
Q(z,w,Z) = cn(1 — (z,w))n^]TCk+k-1Cln+l-1 f(Z)(z, Z)k(Z, w)'da(Z). (8)
_n ;_n ^ S
k=01=0
The integral / f (Z)(z, Z)k (Z, w)1 da(Z) is a homogeneous polynomial of degree of homogeneity k Js
in z and l in w. Multiplying the sum of Eq. (8) by a factor of (1 — (z, w)) and regrouping the terms we find that
o
$(z,w)=^ PM(z,w), (9)
k,1=0
where Pk1(z, w) are homogeneous holomorphic polynomials of degree of homogeneity of k in z and l in w. The double series converges absolutely in B x B and uniformly on any compact set in B x B.
Theorem 3. Let the function f (Z) G C(S), the point a G B and let the function $(z, w) satisfy
da$(0, w)
the conditions: $(0, w) = const, $(a, w) = const, -a- are polynomials in w of degree not
higher than ||a||, then for any fixed z, belonging to the complex line 10,a = {z G Cn : z =
$(z,w)
at, |t| < 1}, function $(z, w) = const in w, i.e., —p— =0 at ||ft|| > 0. Proof. We represent the function $(z, w) in the form (9):
o
$(z,w) = Y^ (z, w).
k,l=0
By the hypothesis expansion of (9) takes the form
$(z,w) = pk,i(z,w),
da+p$(0, 0)
since the derivatives —-—-—3— =0 at ||fl|| > ||a||.
dzadwp 11 ......
o
We introduce the functions $k(z, w) = Pk,i(z,w), then
l=k
o
$(z,w) = £$k (z, w). (10)
k=0
Considered series converges absolutely in B x B, uniformly on compact subsets of B x B, since double series (9) converges absolutely in B x B and uniformly on compact subsets of B x B, and the series (10) is repeated series of series (9).
From the form of the series $k(z, w) we get that $k(tz,w) = tk$k(z,w) for every t G C. By the hypothesis
and
$(0,w) = $0(0,w) = ^ p0,i(0,w) = const (11)
l=0
o
$(a, w) = ^^2 $k(a, w) = const.
k=0
Consider
We calculate
m
o
$(at, w) = ^2 tk$k(a, w). (12)
k=0
d
-—$(at, w) = m!$m(a, w) +-----+ k(k — 1) • ... • (k — m + 1)tk-m$k(a, w) +
dtm
We calculate the derivative of the same as the derivative of a composite function
dm ^ da$(at, w) a
-$(at,w)= > -' 7 • aa
dtm v ' ^ dza
l al =m
Equating the derivatives , we obtain
E -dO^ • a" = E k(k - !) • • • • • (k - m + 1)tk-m$k(a, w). (13)
||a||=m k=m
dm
Substituting value t = 0 in Eq. (13), we get that dm$(0, w) = m!$m(a,w) is a polynomial of degree not higher than m in w, since the left side of this equation is a polynomial of degree not higher than m in w by hypothesis of the theorem. When m = 0 we get $(0, w) = $0(a, w) = const and from (11) we have $(0, w) = $0(a, w) = $0(0, w). Substitute t = 1 in Eq. (12), we obtain
$(a, w) = ^^ $k (a, w) = const.
k=0
Since $k(a,w) = ^^Pk,i(a,w) is a polynomial in w of degree not higher than k, then
l=k
<x>
^^Pk,i(a,w) = Pk,k(a,w). Therefore
i=k
const = &(a,w) = (a,w) = ^^ Pkk (a,w).
k=0 k=0
Hence Pk,k(a,w) = 0 for k > 0. Therefore $k(a,w) =0 for k > 0, so from (12) we get
, . , d3$(at, w) &(at, w) = const and -' ' =0 at H^H > 0. □
dw3
Corollary 2. Let the function f (Z) £ C(S) have the one-dimensional holomorphic extension property along the family L{o,a}, then &(z,w) = const for points z belonging to the complex line
lo,a n B, i.e., ^=0 at H£H > 0-
Proof follows from Theorem 1 and Theorem 3. □
Corollary 3. Under the hypotheses of Corollary 2 the equality
d3 F (z) =0 dz 3
holds for all points z £ l0,a n B and > 0.
Theorem 4. Let the function f (Z) £ C(S) have the one-dimensional holomorphic extension property along the family L{ac}, a,c £ B, then $(a + (c — a)t,w) = const on w for |t| < 1, i.e., d3 ^a +(c — a)t,we ,, ,,
1 dwp ' =0 at llfll > 0.
Proof. From Theorem 2 we have
^ ^ 3a$(a,w) 3a$(c,w)
&(a,w) = const, ®(c, w) = const and ---, ----(14)
dza dza
are polynomials of degree not higher than llall in w.
Expanding the function $(z, w) in a Taylor series at the point (a, a), we obtain
$(z,w)=^ Pk,i(z - a,w — a), (15)
k,l =0
where Pkjl(z,w) are homogeneous holomorphic polynomials of degree of homogeneity of k in (z — a) and l in (w — a). The series (15) converges to the function $(z, w) in the neighborhood of (a, a). We make the substitution z — a ^ z, w — a ^ w. Then the point c go to point c = c — a, and the function $(z, w) will pass to a function $(z, w) holomorphic in a neighborhood Uz x Uw of zero, where Uz is some neighborhood of zero at z, and Uw is some neighborhood of zero at w, and the series (15) will take the form
^
$(z,w) = ^ Pk,l(z,w), (16)
k,l
which converges in the neighborhood Uz x Uw. Therefore, the conditions (14) take the form
, da$(0,w) da$(c,w) , ,
$(0, w) = const, $(c, w)= const and ---, ---. (17)
dza dza
So they are polynomials in w of degree not higher than ||a||. From the conditions of the theorem the expansion (16) will take the form
$(z,w) Pk,i(z,w), (18)
k^l^O
3a+P $ (0,0)
since the derivatives —-—-—3— =0 at ||fl|| > ||a||.
3zadwP 11 ......
We introduce the functions
$k (z,w) = ^ Pk,i(z,w), (19)
l=k
then
$(z,w) = £$k (z, w). (20)
k=0
Considered series (19) and (20) converge absolutely in Uz x Uw, uniformly on compact subsets in this neighborhood, since double series (16) converges absolutely in Uz x Uw and uniformly on compact subsets in this neighborhood. Then
$(0, w) = $0(0,w) = Po,i(0,w) = const.
1=0
Consider the expansion of the function $ (tc, w) in a series by t
$ (ic,w) = ^ tk $ k (c,w), (21)
¡k
k=o
~ dk $ (tc, w)
where $k(c, w) = — '— . If we apply the expansion (20) to the point (c, w), we obtain k!dtk t=0
$ k (c,w) = $k (c,w) (22)
for sufficiently small |w|. We find
dm ~ ~ ~ — $ (tC, w) = m!$m(c, w) + ... + k(k — 1) • ... • (k — m + 1)tk-m$ k(C, w) + ... .
We calculate this derivative of the same as the derivative of a composite function
dm ~ _ . даФ(tc,w)
—Ф (tc,w)= > -' y
dtm y ' ^ dza
\\a\\=rm
ca.
Equating the derivatives we obtain the equality
E d ca = E k(k — 1) • ... • (k — m + 1)tk-m$ k (c, w). (23)
||a||=m k=m
dm ~ ~
Substituting the value t = 0 in Eq. (23), we get that —m$(0, w) = m!$m(c, w) is a polynomial in w of degree not higher than m, because the left side of this equation is a polynomial in w of degree not higher than m from the condition (17). When m = 0 we obtain $(0, w) = $0(c,w) = const and from (21) we have $ (0, w) = $ 0(c,w) = $0(0,w).
From Eqs. (19) and (22) we find that $k(c,w) = Pk,k(c,w) for sufficiently small |w|, i.e., $k (c,w) is a polynomial of degree exactly k in w. Therefore
œ
const = Ф(C, w) = ^^ Pk,k (c, w).
k=0
Hence Pk,k (c,w) =0 at k > 0. Here $(c,w) =0 at k > 0, therefore from (21) we get that
sr, n d3 $ (tc,w)
$(tc,w) = const, i.e., -; ' =0 at HPH > 0.
dw3 „
d3 F (z)
Corollary 4. Under the hypotheses of Theorem 4 the equality
= 0 holds at
z=a+(c-a)t
dz3
lien > 0.
4. Main results
Theorem 5. Let n = 2 and the function f (Z) G C(S) have the one-dimensional holomorphic extension property along the family L{a,c,d}, points a, c, d G B do not lie on one complex line in д3Ф(z w)
C2, then - —-—в— = 0 for any z G B and ||в|| > 0, i.e., F(z) is holomorphic in B (then f (Z) extends holomorphically into B).
Proof. Let z be an arbitrary point on the line la,c, then by the Theorem 4 we have
двФ(С, w)
dwe
(24)
at H^ll > 0. We connect the point 5 to the point d of the line la,d and again apply Theorem 4
„ , . = , , d3&(5,w
for the points z G lz d, we get that —-—3—
dw3
satisfied for all points 5 from some open set.
d 3 $(z,w)
for the points z £ d, we get that —-—— =0 at HpH > 0. Therefore, the condition (24) is
dw3
dp F (z)
Substituting in Eq. (24) w = a and using Eq. (5), we find that ——5— =0 in an open set
dzp dF (z)
of the ball B. Due to the real-analytic of function F(z), it follows that ——— = 0 for any z e B
dzj
and j = 1,..., n. Since F(z)|dB = f (Z), then the function f (Z) extends holomorphically to the ball B. □
From Theorem 5 we get that in the ball B c C2 a sufficient set for a continuous function defined on the boundary of the ball is the set £{a,c,d}, where a, c, d are arbitrary points of the ball, not lying on one complex line.
We denote A the set of points ak e B c Cn, k = 1,..., n+1, do not lie on complex hyperplane in Cn.
Theorem 6. Let f (Z) e C(S) have the one-dimensional holomorphic extension property along dP $(z, w)
the family La, then —^—p— = 0 for any z e B, i.e., F(z) is holomorphic in B (then f(Z) extends holomorphically into B).
Proof. The proof goes by induction on n. The basis of induction is Theorem 5 (n = 2). Suppose that for all dimensions k < n theorem is true. Without loss of generality when k = n, we assume that an+1 = 0.
Consider the complex plane r, passing through the points ai,..., an, its dimension, by assumption, is n — 1 and 0 / r. The intersection of r n B is some ball in Cn-i. The function f |rnS is continuous and has the property of holomorphic extension along the family La, where
dp$(z', w)
A1 = {a1,..., an}. By the induction assumption --—-5-=0 at || > 0 for all z' e r n B.
dwp
dp $(z, w)
Connecting points z' e r with the point 0 we get by Theorem 3, that —-—7.— = 0 at
dwp
||ft|| > 0 for some open set in B. Hence, as in Theorem 5, we see that F(z) is holomorphic in B.
□
dp F (z)
Corollary 5. Under the hypotheses of Theorem 6 the equality —5— = 0 holds for any z e B
dzp
and ||ft|| > 0 and f (Z) extends holomorphically into B.
The authors used the financial support of RFBR, grant 14-01-00544-
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Голоморфное продолжение непрерывных функций вдоль конечных семейств комплексных прямых в шаре
Александр М. Кытманов Симона Г. Мысливец
В этой статье мы рассматриваем непрерывные функции, заданные на границе шара B в Cn, n > 1, и обладающие одномерным свойством голоморфного продолжения вдоль семейств комплексных прямых, проходящих через конечное число точек из B. Мы доказываем, что это семейство является достаточнымдля голоморфного продолжения таких функций в шар B.
Ключевые слова: голоморфное продолжение, ядро Пуассона, комплексные прямые.
