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EDN: RXMMFV УДК 517.55
Green's Function on a Parabolic Analytic Surface
Azimbay S. Sadullaev*
National University of Uzbekistan Tashkent, Uzbekistan
Khursandbek K. Kamolov^
Urgench State University Urgench, Uzbekistan
Received 10.09.2022, received in revised form 16.11.2022, accepted 20.01.2023 Abstract. The class of plurisubharmonic functions on a complex parabolic surface is considered in this paper. The concepts of the Green's function and pluripolar set are also introduced and their potential properties are studied.
Keywords: parabolic manifold, parabolic surface, regular parabolic surface, Green's function, pluripolar set.
Citation: A.S. Sadullaev, Kh.K. Kamolov, Green's Function on a Parabolic Analytic Surface, J. Sib. Fed. Univ. Math. Phys., 2023, 16(2), 253-264. EDN: RXMMFV.
1. Introduction and preliminaries
This paper is devoted to plurisubharmonic (psh) functions on a complex parabolic manifold and surfaces embedded in the space CN. The concepts of Green's function and pluripolar set are introduced and a number of their potential properties are studied.
Parabolic manifolds presumably were considered for the first time by P. Griffiths, J.King [1] and by W. Stoll [2,3]. They were used in the construction of the multidimensional Nevanlinna theory for holomorphic map f : X ^ P, where X is a parabolic manifold dim X = n, and P is a compact Hermitian manifold, dim P = m. Various types of parabolicity were classified by A. Aytuna and A. Sadullaev [4-6].
Definition 1. A Stein manifold X C CN, dim X = n is called parabolic if it does not contain different from a constant plurisubharmonic function bounded from above, i.e., if u(z) is plurisubharmonic on X and u(z) ^ C then u(z) = const.
It is called S-parabolic manifold if it contains a special exhaustion function p(z) that satisfies the following conditions
a) p(z) e psh(X), {p < c}cc X yc e R;
b) (ddcp)n = 0 outside some compact K CC X, i.e., function p is maximal function on X\K.
X is called S* -parabolic if there is a continuous special exhaustion function p(z) on it.
It is clear that S*-parabolic manifold is S-parabolic and, in turn, it is easy to prove that S-parabolic manifold is parabolic. It was noted [5,6] that for n =1 all these 3 concepts coincide (see [7]). However, for n > 1 the equivalence of these three definitions is still an open problem.
* [email protected] https://orcid.org/0000-0003-4188-1732 [email protected] https://orcid.org/0000-0002-4314-7243 © Siberian Federal University. All rights reserved
The purpose of this paper is to study holomorphic and plurisubharmonic functions on an analytic surface. Concepts of parabolic surfaces X c CN, plurisubharmonic functions and Green function on them are introduced (Section 2). A series of properties of plurisubharmonic on X functions are proved. Some of these properties are non-trivial due to the presence of singular (critical) points of X. When proving properties in neighbourhoods of such points, the local principle of analytic covering is used. In Section 3, the concepts of polynomials are defined, the class of polynomials on parabolic surfaces is studied, and a number of examples of surfaces of this type are given. Theorem 3.1 states that complement X = CN\A of an arbitrary pure (n — 1)-dimensional algebraic set A = {p(z) = 0} c CN is regular S*-parabolic surface.
2. Parabolic surfaces
In this section, parabolic surfaces, their classification and the Green's functions on them are we studied.
2.1. Plurisubharmonic functions on analytic surfaces.
Let us consider an analytic surface, i.e., an irreducible analytic set X, dim X = n embedded in a complex space CN, X c CN such that for any ball B(0, r) c CN the intersection X n B(0, r) lies compactly on X, X n B(0, r) cc X. To define plurisubharmonic functions on X we denote the set of regular points of the set X by the X0 c X. Then the set of critical (singular) points X\X0 is an analytic set of lower dimension dim X\X0 < n. Set X\X0 does not split X, and set X0 is a complex n dimensional submanifold in CN (see [8,9]).
Definition 2 ([10]). Function u(z) defined in a domain D c X is called plurisubharmonic (psh) in D if it is locally bounded from above in this domain and plurisubharmonic on the manifold D n X0, u(z) e psh(D n X0).
The class of plurisubharmonic functions in D is denoted by psh(D). In practice, at critical points z e X\X0 function u*(z) = lim u(w), z e D is usually considered, and in studies of
w^z wex°no
plurisubharmonic functions u*(z) is studied. Function u*(z) is assumed to be upper semicontin-uous in D, the set {z e D : u*(z) < C} is open for all C e R and u*(z) = u(z), Vz e X0 n D.
Let us consider several properties of plurisubharmonic functions on X that are needed below.
1) A linear combination of finite number of plurisubharmonic functions in D c X with positive coefficients is a plurisubharmonic function, i.e., if u*(z) e psh(D), aj ^ 0, j = 1, 2,..., s then
a\u*(z) + ... + asu*(z) e psh(D).
2) The uniform limit or the limit of a monotonically decreasing sequence {u*(z)} of plurisubharmonic functions is plurisubharmonic function, i.e., if u*(z) e psh(D), j = 1,2,..., u*(z)^u*(z) or if u*(z) \ u*(z) then u*(z) e psh(D).
The following property is not trivial due to the presence of critical points on the surface X.
3)Maximum principle. For u*(z) e psh(D), D c X the maximum principle holds, i.e., if at some interior point
z0
D the value u*(z0) = supu*(z) then u* = const.
D
Now let us assume that u* has a maximum at an interior point z0 e D (without loss of generality one can assume z0 = 0) and u*(0) ^ u*(z) Vz e D. If 0 e D is regular point then it
is obvious that u(z) = const in D\S, where S = X\X0, because for plurisubharmonic functions on a manifold the maximum principle is valid. Therefore, u*(z) = const in D. If 0 is a critical point then there is a complex plane 0 G n c CN such that dimn = N — n, X n n is discrete. Hence, there exists a ball B(''0,r) c n, r > 0 such that
X n B(''0, r) = {0}, X n dB(''0, r) = 0. (1)
Let us set z = ('z,''z), 'z = (z1,..., zn), ''z = (zn+1,..., zN). Let n = { 'z = (z1,..., zn) = 0 }. Since X is closed then according to (1), there exists a neighbourhood ' U B '0 : Xn dB(' z, r) = 0, V'z G 'U. Therefore, n : X n ['U x ''U] ^ 'U is a k-sheeted analytic covering, 1 ^ k < to. Let J c 'U be the set of critical points of this covering. It means that
n : {X n ['U x ''U]}\n-1(J) ^ 'U\J
is a regular k-sheeted covering
n-1(' z) n {X n [ 'U x ''U]}\n-1(J) = {a1 ('z), ...,ak (' z) } V 'z G 'U \J. (2)
Moreover, for each point' z0 G ' U \J locally, in some neighbourhood of W B 'z0, the inverse-image n-1(W) n X n ['U x ''U] is split into k pieces of disjoint complex manifolds M1, M2, ..., Mk (see, for example, [8,11]). Function u*(z) = u(z) is plurisubharmonic function on every piece of manifolds Mj, j = 1,2,... ,k.
k
It follows from (2) that w('z) = ^ u*(a.j('z)) is plurisubharmonic function in 'U\J locally
j=1
bounded in 'U. Since J c 'U is an analytic set then w('z) is plurisubharmonically extended to 'U. (Recall that if w('z) G psh(D\P) is locally bounded in D, P is closed pluripolar set then w('z) is plurisubharmonically extended to D (see [12] and also [13,14]).
Thus, w( z) G psh( U) and by assumption it reaches its maximum at the point 0 G U. This is a contradiction. □
2.2. Holomorphic functions
It is convenient for us to define holomorphic functions on an analytic surface X in the sense of H.Cartan [9].
Definition 3. Function f (z) defined in a domain D c X is called holomorphic in D, if:
a) it is holomorphic on the manifold D n X0;
b) it is locally bounded in D, i.e., for each point
z0
D there exists a neighborhood W B z ,
W c D such that \f (z)\ < const , Vz G W n X0.
The class of holomorphic functions in D is denoted by O(D). Holomorphic functions in space X have many properties of holomorphic functions of several complex variables. In particular, a linear combination of holomorphic functions with constant coefficients is holomorphic function. In other words, if f1,...,fm G O(D) then c1f1 + ... + cmfm G O(D); the product of two holomorphic functions is also holomorphic function, i.e., if f,g G O(D) then f ■ g G O(D). In addition, the theorem of uniqueness holds, i.e., if f G O(D) and f = 0 in some non-empty neighbourhood of W c D then f = 0 in domain D c X. Since holomorphic functions are defined only at regular points then f = 0 in some neighbourhood W c X means that f = 0 Vz G WnX0. The following theorem of H. Cartan is very useful in the study of holomorphic functions.
Theorem 2.1 (H. Cartan [9]). Let function f (z) defined in a domain D c X is continuous on D n X0 and has the property that for each point z0 e D there exist a neighbourhood W B z0, W c D, and holomorphic in W functions gi,..., gm e O(W) : fm(z) + gi(z)fm-1(z) +... + gm(z) = 0 Vz e W n X0. Then f (z) eO(D).
There is an intimate connection between holomorphic and plurisubharmonic functions.
Theorem 2.2. If f (z) e O(D) then function u(z) = ln \f (z)\ is plurisubharmonic in domain D, u(z) e psh(D).
2.3. ¿"-parabolic analytic surfaces
Let X c CN be an analytic surface embedded in CN, i.e., X is an irreducible analytic set in CN for which the intersections B(0,r) n X cc X, Vr > 0. The concept of parabolicity of surface X is introduced similarly to the parabolicity of manifolds.
Definition 4. An analytic surface X is called parabolic if it does not contain a bounded plurisubharmonic function that is different from a constant.
Analytic surface X is called S-parabolic if it has a special exhaustion function p(z) satisfying the following conditions
a) p(z) e psh(X), {p < c} cc X Vc e R;
b) function p* is a maximal function on X\K for some compact set K cc X. This is equivalent to (ddcp*)n = 0 on X0\k (see [15]).
Analytic surface X is called S* -parabolic if there exists a continuous special exhaustion function p(z) e C(X0).
It is clear that S*-parabolic analytic surface is S-parabolic. As we noted above, the converse assumption remains open even for a complex manifold. The main result of Section 2 is the following theorem.
Theorem 2.3. S-parabolic surface X is parabolic, i.e., on the S-parabolic surface X there is no bounded from above plurisubharmonic function u*(z) different from a constant.
Proof. Let X be a S-parabolic analytic surface with special exhaustion function p(z) e psh(X), and p is a maximal plurisubharmonic function on X\K, where K cc X is some compact set. Suppose that there exists function u(z) e psh(X), u(z) < M but u(z) ^ const. Consider a ball Br = {z e X : p(z) < ln r} cc X. Let us put pr = max p(z), ur = max u* (z), ur ^ M. Let us
Br Br
fix the numbers r < r' < R < to, Br D K. Then for P-measure (see [15]) we have
P(z) — P R PR — P
u*(z,Br ,Br)= ' ' ■ (3)
Let us note that u* (z) ^ ur, z e Br and u*(z) ^ uR, z e BR. Therefore, by the theorem on two constants [15] we have
u* (z) ^ ur ■ (1 + M*(z, Br, Br)) — ur ■ u*(z, Br, Br).
Substituting (3) into the last inequality, we obtain for z e Br
^ f, . Pr' — PR) Pr' — PR
ur' ^ 1 +--\ur--ur .
V PR — Pr J PR — Pr
Since function u(z) is bounded on X then uR ^ M, and when R ^ to we have ur' ^ ur. Hence, according to the maximum principle, u*(z) = const in the ball Br. Since r < to is an arbitrary fixed number then u*(z) = const on X. Theorem 2.3 is proved. □
2.4. Green's function on parabolic surfaces
In this subsection the Green's function on S-parabolic analytic surfaces is introduced. Let (X, p) be a S-parabolic surface. Let us denote the class of plurisubharmonic functions u G psh(X) satisfying the condition
u(z) ^ cu + p+(z), z G X,
by the Ap(X). Here cu is some constant that depends on function u and p+(z) = max {0, p(z)}. Class A p (X) is called the Lelong class of plurisubharmonic functions on X. For a fixed compact set K cc X, we define
Vp(z, K) = sup{u(z) : u G Ap(X), u\k < 0}.
Then the regularization V**(z,K) = lim Vp(w, K) is called p-Green's function of the compact
p w^z
k cc X.
Similarly, in the classical case there are
1. Either Vp G Ap (X) or Vp = +to. Vp(z, K) = +to if and only if K is pluripolar set on X, i.e., there exists a function u* G psh(X) : u* ^ —to, u*(z) = —tovz G K.
2. Let K cc X be a non-pluripolar compact set. Then the Green's function Vp(z,K) is maximal in X\K. In particular, [ddcVp(z, K)]n = 0 on the complex manifold X0\K.
The proofs of these important properties of the p— Green's function are identical to the proofs of the corresponding properties of the Green's function in space Cn, and they are omitted.
Definition 5. A compact set K c X is called regular at a point z0 G X if V*(z°, K) = 0. If all points of K c X are regular then compact K c X is called regular.
Note that if compact set Kc X is regular then the open set Ge = {z G X : V*(z, K) <e} contains K, Ge D K.
2.5. Regular parabolic surfaces
2.5.1. Polynomials on parabolic analytic surfaces
Let X c CN be a S-parabolic surface and p(z) is a special exhaustion function.
Definition 6. If function f G O(X) satisfies the inequality
ln\f(z)\ < dp+(z) + Cf Vz G X, (4)
where cf and d are positive real numbers (constant) then f is called the p-polynomial. The smallest value d that satisfies condition (4) is called the degree of polynomial f.
Let us denote the set of all ^-polynomials of degree less than or equal to d by Vd(X) and the
union Pp(X) = y Pd(X) by Pp(X). Then it is easy to prove (see [6,16]) that Pdp(X) is a linear
space of finite dimension dimPp(X) ^ C (d + 1)n.
However, a parabolic manifold was constructed [6] where there are no non-trivial polynomials, i.e., any polynomial P(z) on X is equal to a constant, Pp ~ C.
Definition 7. If space of all p-polynomials Pp(X) = P^(X) is dense in space O(X) then
d^l
S-parabolic surface X is called regular.
2.6. Examples
Example 1. Let A c CN be irreducible, n dimensional, dim A = n, n < N algebraic set. According to the well-known criterion of W.Rudin [17] (see also [18]) and after corresponding linear transformation, algebraic set A can be included in a special cone
A c {w = (' w,'' w) = (wi,..., wn,wn+1,..., wn ) : ||''w|| < C (1 + || 'w||)} ,
where C is constant.
Let us consider projection n('w,''w) = ' w : A ^ Cn. If ('w0,''w0) is a regular point of A, i.e., ('w0, ''w0) e A0 then in some neighbourhood U B ('w0, ''w0), U c A0 projection n : U ^ Cn is biholomorphic. Consequently, restriction p\A of the plurisubharmonic in CN function p(w) = ln ||' wH is plurisubharmonic function in a neighbourhood of U B ('w°,''w°). Since point (' w0,''w0) e A0 is arbitrary restriction of p\A is a plurisubharmonic function in A0. In addition, it is locally bounded from above on A and, therefore, p\A e psh(A).
It is clear that p\A is special exhaustion function on A and restriction on A of polynomials p(' w,'' w) from CN are polynomials on A. This implies that set of polynomials Pp(A) is dense in O(A), i.e., affine-algebraic surface is regular parabolic surface.
Example 2. Let A = {&(z) = 0} c Cn be a pure (n — 1) dimensional analytic surface such that
A c{z = ('z, zn) e Cn : \zn\ < y('z) },
where ' z = (z1,..., zn-1), 'z) is a locally bounded positive function. Then A is S* -parabolic surface.
Let us consider projection n('z,zn) = ' z : A ^ Cn-i. For each fixed point 'z0 e Cn-i intersection {'z = 'z0} n A = n-1{'z0} consists of a finite number of points {'z = 'z0} n A = (a1('z0),..., am('z°)) as a compact analytic set in plane C'zo. Function $(' z, zn) = 0 on the boundary of circle {\zn\ = y('z)}. According to the argument principle, the number of zeros (taking into account multiplicities)
1 r & ('z, zn) ,
' ■ dz.
N('z) = n ! dzn, z G U,
2m J 'z, zn)
\zn\=v('z°)
as a continuous integer function is constant, N(' z) = m and n-1('z) = (a1('z),...,am('z)), 'z e Cn-i. Moreover, function
m
F (' z, zn) = W(zn — ak (z)) = zm + fm-i(' z)zm-1 + ... + f0(' z) k=1
is an entire function, where fk('z) e O(Cn-i), k = 0,1,... ,m — 1.
If A is not an algebraic set then function F(' z, zn) is not a polynomial, i.e., not all functions fk (' z) e O(Cn-i), k = 0,1,... ,m — 1 are polynomials. As in Example 1, contraction p\a of plurisubharmonic function p(z) = ln ||'zH from CN is special exhaustion function on A, i.e., surface A is parabolic. However, here restrictions of polynomials P(z) in Cn on A are not, in general, p\A polynomials.
It was proved ([5], see also [19]) that X = Cn\A complement of zeros of the Weierstrass polynomial A = {zm + f1(' z)z"m-x + ... + fm(' z) = 0}, where f1(' z),..., fm(' z) are entire
functions, is S* -parabolic manifold with special exhaustion function
F (z) +
p(z) = \\J \z\2 +
F(z) ) '
where F(' z, zn) = zm + fm-1('z)z~m + ... + f0(' z). However, X = Cn\A is not always regular (see [19]). The main result of this section is
Theorem 2.4. The complement X = Cn\A of an arbitrary pure (n — 1) dimensional algebraic set A = {p (z) = 0} c Cn is regular S* -parabolic manifold. If p (0) = 0 then function
p (z) = — ^grp In \p (z)\ + 2ln ||z|| (5)
is special exhaustion function on X.
The theorem is proved in several steps.
Step 1. Let us show that p (z) from (5) is special exhaustion function. In fact, function ln \p (z)\ is pluriharmonic in X and function 2\n ||z|| is maximal in X\{0}. Therefore,
deg p
function p (z) is the maximal function in X\{0}. In addition, since p (0) = 0 then {z G X : p(z) <C}cc X VC> 0.
Step 2. Using the criterion of W.Rudin [17] and after the corresponding linear transformation of space Cn, A is reduced into special form (see Example 1)
A c{z = ('z,zn) G Cn : \zn \ <C (1 + ||'z||)} , C — const. (6)
Then A has the form
A = {p(z) = zm + e1 (' z)zm-1 + ... + em (' z) = 0} ,
where m = degp > 1, e1('z),..., em('z) are polynomials and p(0) = 0.
Step 3. The expansion of holomorphic functions in X = Cn \A into Jacobi-Hartogs series is used. First, let us consider some insights on the theory of Jacobi series ([20], see also [21]). Let p(z) = zm + e1zm-1 +... + em, m > 1 and e1,...,em are constants. Let us denote the lemniscate ring {z G C : r < \p (z)\ < R} by Gr,R. If function f (z) is holomorphic in some neighbourhood Gr,R then function of two variables
F ^^ f ft- «
2m J p(£) — w £ — z
dOr,R
is holomorphic in domain Gr,R x {r < \w\ < R}. According to the Cauchy integral formula, the equality F(z,p(z)) = f (z) (z G Gr,R) takes place. The expansion of function F(z,w) into Hartogs-Laurent series (see [11]) with respect to the variable w is
(z)wk
F(z,w)= Y, ck(z)wk, (7)
k = -*x>
L f f(n p(t) - p(z)
J f (i;)pk+1(^)(^ - z) (8)
(r<r1 < R, z e Gr,R, k = 0, ±1, ±2,...).
k = -
where
Ck(z) = ^ I f dC
Series (7) converges uniformly inside domain Gr,R x {r < |w| < R}. If we put w = p(z) then we obtain the series
f (z) = Y^ ck(z)pk(z), z G Cr,R,
k = -
oo
which is called the Jacobi-Hartogs series of function f (z). It converges uniformly inside domain Gr,R. One can be see from (8) that coefficients ck (z) are polynomials of degree deg ck (z) < m — 1. It follows that if function f (z) is holomorphic in G0,œ then it is expanded into the series
f(z)= J2 Ck(z)pk(z),
k=
which converges uniformly inside Gq,^. Here ck(z) are polynomials of degree deg ck(z) ^ m — 1,
Ck(z) = 2ni J f ® pk%t-z) * (0 <r< G G0^,
and the Cauchy inequality holds:
f P(0 - P(z)
( < max [\f (g)| : \p (g)| = r} f
\Ck(z)| <-2nrk+i- J
\p(£)\=r
e - z
\di\, k = 0, ±1, ±2,... (9)
Step 4. Let us apply the Jacobi-Hartogs series to the holomorphic function f ('z, zn) G O (X) outside the algebraic set A = {p (z) = zm + e1 ('z) zm-1 + ... + em ('z) = 0}. We fix 'z G Cn-1 and expand function f (' z, zn) in the Jacobi-Hartogs-Laurent series:
f ( 'z,zn)= Y Ck ( 'z,zn ) ■ Pk ( 'z,zn) (10)
k=-œ
where coefficients
1 i ît, ^ P('z,in) - P('z,zn) „
-din
1
ck( z,zn) = 2ni f ( z,in) ' : 7"Sn
Pk+1(' z,£n)(£n — zn)
\p('z,În)\=r
are polynomials in variable zn with holomorphic Cn-1 coefficients
Ck ( 'z, zn) = ak,m-i ( ' z) zm-1 + ... + ak,0 ( ' z), ak,j ( ' z) G O (Cn-1) , j = 0, 1,...,m — 1. Series (10) converges uniformly inside domain
X = {('z,zn) G Cn : 0 < \p ('z,zn)l < œ} .
Step 5. Let us show that rational functions of the form qM\ , where q (z) is a polynomial
pk (z)
in Cn, k ^ 0 are integer functions, and only they are ^-polynomials in X, where p(z) = =
---ln \p (z)\ + 2ln ||z||. In fact, since
deg p
q (z)
ln
pk (z)
= —k ln \p (z) \ + ln \q (z)\ ^ max {k, deg q} p+ (z) + const
then function ^^ \ is p (z)-polynomial in X = Cn\A.
pk (z)
On the contrary, if P(z) is a p(z)-polynomial in X = Cn\A, then according to (4) ln \P(z) \ < dp+ (z) + c V z G X, d,c — const. Let us expand P(z) into the Jacobi-Hartogs-Laurent series (10)
f ('z,zn)= ^ ck ('z,zn) ■ pk ('z,zn),
k=-w
with coefficients
Ck ('Z,Zn) = 2~ f P('z>£n)
\p{'z,£n)\=r
According to (9), we have the following estimate
max {\P ('z,£n)\ : \p ('z,£n)\ = r}
p('Z,Cn) - p('z,zn)
d£n-
\ck (' Z,Zn)l <
Pk + 1('Z,Cn)(Cn - Zn)
f P ('Z,Cn) - P ('Z, Zn)
2nrk+1
<
max {exp [c + dp+ ('z,Cn)] : \p ('z,Cn)\ = r} 2nrk+1
\p('z,În)\-
\dCn\ < №n\.
Substituting p(z) = — d-In \p (z)\ + 2\n ||z||, we obtain
\ck ('Z, Zn)\ <
{exp [c + d (-m ln \P ('Z,in)\ + 2ln ||('z,£n)||)+] : \p ('z,Cn)\ = r}
x J
\P( ' z,in)\=r
However,
p (' Z,Cn) - p (' Z,Zn)
2nrk+1
Cn Zn
dn\.
(ii)
ln \p (z)\ + 2ln ||z||
max < exp c + d I - — 1 [ ^ d
c + d( - m +ln l(' ^ )llP' z,U\=r)
^ exp ^ exp Here the estimate
: \p ('z,Cn)\ = r\ <
c + d ( - — + ln (r2 + C1 m V
'zH'
(12)
< C2>
! + ll'zfj if r — œ fd ■ r-d/m if r — 0
z,tn) =r
<
ll 'zll
2 , |2
< ll 'zll2 + r2 + C2 (1 + ll 'zllf,
\P( ' z,in )|=r
is used which is easy to obtain by applying relation (6). The integral in (11) is estimated as (see [19] Lemma 4.1)
\p( ' Z,£n)\=r
p ('Z,Cn) - p ('Z,Zn)
Cn zn
\dCn\ < Csr, \p ('Z, Zn) \ < r, C3 - const.
(13)
r
r
max
x
+
+
Thus, substituting (12), (13) into (11), we obtain the final estimate
d
, ,, Ca \ (r2 + II 'z\\ ) if r —y TO
\ck ('z,zn)l < { V 11 ,k = 0, ±1, ±2,...,Ca — const. (14)
r'k 1 "' \\2d ■ r-d/m if r — 0 For indices k > 0 the upper inequality (14)is used:
\ck ('z,zn)\ < -k (r2 + \\'z\\2^ — 0, for r — to and k > 2d. For indices k < 0 the lower inequality (14) is taken:
—3 2d d
\ck (' z, zn) \ < —j- \\'z\\ ■ r-d/m — 0, with r — 0 and k <--. Consequently, ck (' z, zn) = 0 for
rk m
all \ k\ > 2d and
+ 2d
f ('z,zn)= ^ ck ('z, zn) ■ pk ('z,zn).
k=-2d
However, according to (14), each function ck(' z, zn), \k\ < 2d, is a polynomial, i.e.,
+ 2d
c' ( ^ zn) f V ~J 2d/l \ '
p2a('z, zn ) k=-2d n
Step 6. It remains to show that space of polynomials Pp(X) is dense in space O(X), i.e., an arbitrary holomorphic function f (z) is uniformly approximated by ^-polynomials inside X = Cn\A This follows from the fact that, as we noted above (step 4), the Jacobi-Hartogs-
Laurent series f ('z,zn) = ^ ck(' z,zn) ■ pk('z,zn) of an arbitrary function f ('z,zn) e O(X)
k=-w
converges uniformly inside X = Cn\A Here coefficients are
f ('z,zn)= ck('z,zn) ■ pk('z,zn)= q( z,zn)
ck (' z, zn) = ak,m-i(' z)zm 1 + ... + ak,o(' z), ak,j (' z) e O(Cn 1), j =0, 1,...,m — 1.
M
Consequently, the partial sums of SM(' z,zn)= ^ ck('z,zn) ■ pk(' z,zn) converge uniformly
k=-M
inside X = Cn\A Approximating coefficients akj (' z) e O(Cn-1), k = 0, ±1,..., ±M, j = 1,2,...,m — 1 by polynomials, we thereby obtain approximation of function f ('z,zn) e O(X) by polynomials, i.e., Pp(X) = O(X). Theorem is proved. □
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Функция Грина на параболической аналитической поверхности
Азимбай О. Садуллаев
Национальный университет Узбекистана Ташкент, Узбекистан
Хурсандбек К.Камолов
Ургенчский государственный университет Ургенч, Узбекистан
Аннотация. В данной работе рассматривается класс плюрисубгармонических функций на комплексной параболической поверхности, вводятся понятия функции Грина и плюриполярных множеств, изучается ряд их потенциальных свойств.
Ключевые слова: параболическое многообразие, параболические поверхности, регулярные параболические поверхности, функции Грина, плюриполярные множества.
