DELTA-EXTREMAL FUNCTIONS IN CN Текст научной статьи по специальности «Математика»

DOI: 10.17516/1997-1397-2021-14-3-389-398 УДК 517.55

Delta-extremal Functions in Cn

Nurbek Kh. Narzillaev*

National University of Uzbekistan Tashkent, Uzbekistan

Received 28.01.2021, received in revised form 01.03.2021, accepted 25.04.2021 Abstract. The article is devoted to properties of a weighted Green function. We study the (S, ф)-extremal Green function Vg (z, К,ф) defined by the class Ls = {u(z) € psh(Cn) : u(z) < Cu + S ln+ \z\, z € C"}, S > 0. We see that the notion of regularity of points with respect to different numbers S differ from each other. Nevertheless, we prove that if a compact set К С C" is regular, then S-extremal function is continuous in the whole space C".

Keywords: plurisubharmonic function, Green function, weighted Green function, S-extremal function.

Citation: N.Kh. Narzillaev, S-extremal Functions in C", J. Sib. Fed. Univ. Math. Phys., 2021, 14(3), 389-398. DOI: 10.17516/1997-1397-2021-14-3-389-398.

1. Introduction and preliminaries

The Green function in the multidimensional complex space Cn is one of the main objects for the study of analytic and plurisubharmonic (psh) functions. The Green function was introduced and applied in the works of P. Lelong, J. Sichak, V. Zaharyuta, A. Zeriahi, A. Sadullaev and others (see [1-7]). Recall that a function u(z) e psh(Cn) is said to be of logarithmic growth if there is a constant Cu such that

u(z) < Cu + ln+ \z\, z e Cn,

where ln+ \z\ = max{ln \z\, 0}. The family of all such functions is called the Lelong class and denoted by L. We also introduce a class L+ as follows:

L+ := {u(z) e psh(Cn), cu +ln+ \z\ < u(z) < Cu + ln+ \z\}.

For a fixed compact set K c Cn we put

V(z, K) = sup{u(z) : u(z) e L,u(z)\K < 0}.

Then the regularization of

V* (z,K) = lim V(w,K)

is called the Green function of the compact set K. For a non-pluripolar compact set K, the function V*(z,K) exists (V*(z,K) ^ and belongs to the class L+. The Green function V* (z, K) = if and only if K is pluripolar.

*n.narzillaev<8nuu.uz https://orcid.org/0000-0002-3175-5516 © Siberian Federal University. All rights reserved

Definition 1. A compact set K C Cn is called globally pluri-regular at a point z0 if V*(z°,K) = 0. It is called locally pluri-regular at a point z0 if V*(z0,K n B(z°,r)) = 0 for any ball B(z0, r)), r > 0. A compact set K is globally pluri-regular if it is globally pluri-regular at every point of itself. A compact set K is locally pluri-regular if it is locally pluri-regular at every point of itself.

Theorem 1.1 (see for example, J. Siciak [4], V. Zakharyuta [3]). If a compact set K is globally pluriregular, then the function V*(z, K) is continuous in Cn, and V*(z, K) = V(z, K).

2. Weighted Green functions in Cn

Let '(z) be a bounded function on a compact set K C Cn. Consider the class of functions

L(K,') := {u(z) eL, u(z)\k < '(z)}

and

V(z, K, ') := sup{u(z) : u(z) £ L(K, ')}, z £ Cn.

Then V*(z, K, ') = lim V(w, K, ') is said to be a weighted Green function of K with respect to '^(z). Note that in the case ^(z) = 0 the function V*(z, K, ^) coincides with the Green function V*(z, K), i.e., V*(z, K, 0) = V*(z, K,'). Extremal weighted Green functions are the subject of study by many authors (see [7,10-13]). They are successfully applied in multidimensional complex analysis, in the approximation theory of functions, in multidimensional complex dynamical systems etc.

It is clear that for any compact set K C Cn we have the inequality

V*(z, K) + mKn '(z) < V* (z,K,') < V*(z,K) + max '(z). (1)

If a function '(z) extends to the space Cn as a function from the class L, i.e. if there is a function

^ £ L : ^\k = (2)

then it is obvious V(z, K, ') > ^(z) and

V(z,K,') = '(z) Vz e K. (3)

However, if the condition (2) is not met, then generally speaking, the equality (3) is not true. Example 1. Let K = {\z\ ^ 1} C C and '(z) = 1 — \z\2. Then by the maximum principle

V(z, K, ') = V(z, K) = V(z, K) = ln+ \z\.

Therefore, V(z, K,')=0 < '(z) V\ z \< 1.

According to this example, in order to introduce the concept of regularity, below we assume that the Green function satisfies the condition (3).

Definition 2. We say that a compact set K is globally regular at z0 if V* (z0, K, ') = '(z0). We say that a compact set K is locally regular at z0 if V*(z0,K n B(z°,r),') = '(z0) for every ball B(z0,r), r > 0.

A. Sadullaev [7] proved the following theorem.

Theorem 2.1. Let K be a compact set, and z) is a weight on K such that there exists a strictly plurisubharmonic function

Then K is locally regular at z0 e K if and only if K is globally regular at z .

Note that Theorem 2.1, generally speaking, is not true if ^ is not a strictly plurisubharmonic function. For the weight function ^(z) = 0 and for the compact set K = {\z\ = 1}U{z = 0} c C the point z = 0 is globally regular, but it is not locally regular. In this example K is not polynomially convex K = K. In the work [5] A. Sadullaev constructed the following interesting example.

Example 2. The compact set K = Ki U K2 C C2(zi, z2), where Ki = {\zi\ < 1, z2 = 0}, K2 = = {zi = eip, Rez2 =0, 0 ^ Imz2 ^ ecos v-1, —n ^ <p ^ n}, has the following properties:

a) K is polynomially convex, i.e., K = K;

b) K is globally pluri-regular, i.e., V*(z, K) = 0, Vz e K ;

c) K is not locally pluri-regular at the points z e Ki.

In connection with this example and with Theorem 2.1, the following problem arises (see [7]).

Problem 1. Let K be a compact set in Cn. Under a weaker condition that the weight function z) continues only to a neighbourhood U D K as a strictly plurisubharmonic function, prove that K is locally regular at z0 e K if and only if K is globally regular at z0 e K.

The following theorem relates to local regularity for different weight functions.

Theorem 2.2. Let K be a compact set, and z) is a weight on K : $(z) e C(K). Then K is locally ^-regular at z0 e K if and only if K is locally regular (case ^ = 0) at z0.

Proof. Indeed, we use the inequality (1). If the point z0 e K is not locally pluri-regular, i.e., if

V* (z0, K n B)= a > 0 for some neighborhood B : z0 e B C Cn, then V*(z0, K n Bi) > a for any z0 e Bi c B. Therefore, by (1)

Since ^(z) is continuous, choosing the neighborhood B small enough we can make the right part of (5) to be greater than z0) i.e., V*(z,Kn> ^(z0) and the point z0 is not locally ^-regular.

Reversing the roles of V*(z, K n Bi, ^) and V*(z, K n B\) from (1) we can prove the second part of the theorem: if the point z0 G K is not locally ^-regular, then it is not locally pluri-regular. □

It should be noted here that the conditions of continuity of the function ^(z) in Theorem 2.2 is essential. An example is given in [15], when the function ^(z) is discontinuous, Theorem 2.2 is false, i.e., some point z0 G K c C is a ^-regular point, but it is not pluri-regular.

V GCn C2(Cn) : ddcV > 0, V\K = 4>.

(4)

V*(z°, K n B1, > V* (z0,K n B1) + min ^(z) > a + min ^(z).

(5)

3. ^-extremal functions

Let K C Cn be a compact set and ' (z) be some bounded function on K. Consider the following generalization of the Lelong class

Ls := {u(z) e psh(Cn): u(z) < Cu + 5ln+ \z\, z e Cn}, 6> 0.

It is clear that if v(z) e L, then c • v(z) e Ls, where 0 < c ^ 5. Put

Ls(K,') := {u(z) e Ls, u(z)\k < '(z)}.

Definition 3. The function V*(z,K,') = lim Vs (w,K,') is called a 5-extremal function of K

w^z

with respect to '(z), where

Vs(z, K, ') := sup{u(z) : u(z) e Ls(K, ')}, z e Cn.

We list simple properties of 5-extremal functions:

1°. If 5i < 52, then Vs, (z,K,') < Vs2 (z,K,').

22°. If '1 < '2, Vz e K, then Vs (z,K,'1) < Vs(z,K,'2).

3°. Vs(z,K,') = 5V(z,K, f), in particular Vs(z,K) = 5V(z,K).

4°. Vs(z,K+ c) = c + Vs(z,K,'), Vc e R.

If a function '(z) extends to the space Cn as a function from the class Ls, i.e. if there is a function

^ eLs : *\k = (6)

then it is obvious Vs (z, K, ') > ^(z) and

Vs (z,K,') = '(z) Vz e K. (7)

However, if the condition (6) is not met, then generally speaking, the equality (7) is not true. In this section, as above we assume that the Green function Vs(z,K,') satisfies the condition (7). For such a function ' we can introduce the concept of (5,')-regularity.

Definition 4. We say that a compact set K is globally (5,')-regular at z0 if V*(z0, K,') = = '(z0). We say that a compact set K is locally (5, ')-regular at z0 if V*(z°, K n B(z0, r), ') = = '(z0) for any ball B(z0, r), r > 0.

The following theorem is proved similarly to the proof of Theorem 2.2 and we omit it.

Theorem 3.1. Let K be a compact set and '(z) is a weight on K : '(z) e C(K), Vs(z, K, ') = = '(z) Vz e K. Then K is locally (5, ')-regular at z0 e K if and only if K is locally (5, 0)-regular at z0.

Similarly to Theorem 1.1 the continuity of the 5-extremal function takes place.

Theorem 3.2. Let '(z) be continuous on K. If K is globally (5,')-regular i.e. if K is globally (5, ')-regular at a point z0 e K, then V* (z,K,') = Vs (z,K,') and V*(z,K,') is continuous

in Cn.

Proof. Let 0(z) be a function defined and continuous on K. It is well known that 0(z) can be extended continuously to K, i.e., there is a function ^(z) e C(Cn) such that ^(z)\K = 0(z) (see Whitney H. [8]). We use the standard approximation uj I Vg(z, K, 0), where uj e LgnC^(Cn). Since Vs*(z,K,0) = ^(z), z e K, for any e > 0 there is an open set {z e Cn, Vs*(z,K,0) < *(z) + e} contained K. Therefore, by the Hartogs lemma, there exists j0 e N such that uj (z) < ty(z) + 2e = 0(z) + 2e, Vz e K, j > j0. From here, Uj — 2e e Lg(0, K) and

uj — 2e < V (z, K, 0) < Vs*(z, K, 0) < uj, j > j0, z e Cn.

This means that the sequence uj converges to V$(z,K,0) uniformly and V$(z,K,0) = = V(z,K,0) e c(Cn). □

In the case when S = 1 and 0(z) continues throughout Cn as a continuous function of the class L, Theorem 3.2 was proved by A. Sadullaev.

4. S-extremal functions for different S

Note that in the general case Vg(z, K, 0) and the weight function 0 do not have to be equal on K for all S. In other words, the condition (7) may not be satisfied.

Example 3 (see Alan [10]). Let K = B(0,1) and 0(z) = \z\2. Then one can prove that

, \z\2, \z\ ^Jl,

Vg (z,K,0)={ r r r V2

s s

sln N + ö -ö N >\h-

2 2

We see Vg(z,K,0) = \z\2, Vz e |\z\ ^and Vg(z,K,0) < \z\2, Vz e |^ < \z\ < ^ . We denote by A = A(K, 0) the set of numbers S for which the equality of type (7) holds, i.e.

A = A(K, 0) = {S> 0 : Vg(z, K, 0)\K = 0(z)}. For Alan's example, A = [2, +ro). In fact,

[2ln \z\ + 1, \ z \> 1.

So, V2(z, K, 0)\K = 0(z) and by property 1° from Section 3 Vg(z, K, 0) > V2(z, K, 0) for all S e [2, +x). If S e (0, 2) then there is a point z0 e K such that Vg(zo,K,0) < 0(z0), that is (0,2) n A = 0.

The sets A may be empty. For example, for K = {\z\ < 1} c C and 0(z) = 1 — \z\2, by property 3° we have

Vg(z, K, 0) = Vg (z,K) = SV(z,K) = S ln+ \ z \ .

Therefore, for any S > 0, Vg(z, K, 0) < 0(z), V\z\ < 1. That is, in this case A = 0.

If 0(z) = c, where c is a constant, then Vg(z, K,c) = c + Vg(z, K) = c + SV(z, K). Since the Green function V(z, K) ^ 0,, for any S > 0 and z e K the equality Vg(z, K,c) = c holds. This means that A = (0, +ro).

Let A = 0. If S e A, then from property 1° we easily get Si e A for Si > S. On the other hand

Proposition 1. If Sj e A, Vj e N and Sj I S0 = 0 as j ^ to then S0 e A.

Proof. Indeed, by the hypothesis we have Vgj(z,K,') = '(z),z e K. Using properties 2° and 3°, we get

Vj z K ') = SjVz K j < SjV(z, K f).

Consequently, Vj e N we have '(z) = Vgj (z, K, ') < SjV{z, K, f), z e K. As j tends to infinity, we get

'(z) < S0V(z,K, f) = Vgo (z,K,'), z e K,

i.e. '(z) = S0V(z,K, f) = Vg0 (z,K,'), z e K and S0 e A. □

Proposition 1 follows, if A = 0 then A = (0, to) or A = [S0, +to),S0 > 0. Note that if

S e A(K,'), then Vg(z,K,') = '(z), z e K. Therefore, by monotonicity Vg(z,K n B,') = '(z), z e K n B, for any ball B n K = 0. It follows that if S e A(K, '), then S e A(K n B, ').

Definition 5. Let S e A(K). A compact set K is called globally (S, ')-regular at a point z0 e K if V*(z°, K, ') = '(z0). It is called locally (S, ') -regular at a point z0 e K if for every nonempty ball B(z° ,r) : V*(z°,K n B(z0,r),') = '(z0). A compact set K is globally (S,')-regular if it is globally (S,')-regular at every point of itself. A compact K is locally (S,')-regular if it is locally (S, ')-regular at every point of itself.

Note that global or local (S, ^-regularity can only be defined for S e A. It is easy to see that any locally (S, ')-regular point is globally (S, ')-regular. We denote by Areg = Areg (K,') the set of numbers S c A, for which K is globally regular, we denote by Alrogg = Alrogg(K,') the set of numbers S c A, for which K is locally regular. We see, Alrogg C Areg c A.

Proposition 2. Let Si,S2 e A and Si ^ S2. If a point z0 is (S2,')-regular, then it is (Si,')-regular.

The proof follows from property 1° of Section 3. For a continuous function ' there holds

Theorem 4.1. Let S e A, and a function '(z) be continuous on K. Then a fixed point z0 e K c Cn is locally (S, ')-regular if and only if it is locally pluri-regular.

Proof. We show that for any compact set K c Cn the following is true:

SV *(z,K ) + min '(z) < Vg* (z,K,') < SV *(z,K )+max '(z). (8)

In fact, if u e Lg(K,'), i.e., u e Lg, u\K ^ then

u(z) — max '(z) e Lg(K).

Therefore

u(z) - max ^(z) < Vs*(z,K)

and

V**(z,K,') — max '(z) < V**(z,K) = SV *(z,K), Vz e Cn. Conversely, if u e Lg(K), then u(z) + min'(z) e Lg(K,'). Therefore,

V*(z,K) + min '(z) = SV*(z,K)+min '(z) < V*(z,K,'),

so that (8) holds.

Using (8) we can now prove the theorem. If a fixed point z° e K is not locally pluri-regular, i.e., if V*(z0,KnB) = a > 0 for some neighborhood B : z0 e B C Cn, then V*(z0,KnB1) > a for any z0 e B1 C B. Therefore, by (8)

V**(z0,K n Bl,0) > ¿V* (z0,K n B1) + min 0(z) > 5a + min 0(z). (9)

KnBi KnBi

Since 0(z) is continuous, choosing a neighborhood B1 small enough we can make the right part of (9) to be greater than 0(z0) i.e., V* (z,K n B1,0) > 0(z0). This means that the point z0 is not locally (5,0)-regular.

Reversing the roles of V*(z, K n B1,0) and V*(z, K n B1) from (8) we can prove the second part of the theorem: if a point z0 e K is not locally (5,0)-regular, then it is not locally pluri-regular. □

Corollary 1. Let 51,52 e A and a function 0(z) be continuous on K. Then a fixed point z0 e K C Cn is locally (51,0)-regular if and only if it is locally (¿2,0)-regular.

Proposition 3. If 5j e Areg, yj e N and 5j f 5 as j ^ to, then 5 e Areg. Proof. In fact, since 0(z) = (z, K,0),z e K, we get

0(z) = V* (z, K, 0) = 5j V* (z,K, f) > 5j V* (z,K, f).

Therefore, yj e N we have 0(z) > 5j V*(z, K, f), z e K. As j tends to infinity, we get

0(z) > 5V * (z,K, f) = V*(z,K,0), z e K.

This means that S G Areg.

Corollary 2. If A = [50, to), then Areg =

Corollary 3. If A = (0, to), then Areg =

\ or [So, Si] lor [So, to).

or (0,Si] or (0, to).

In the paper [10] M.Alan studied the concepts of (5,0)-regularity and posed the following problem

Problem 2 ([10]). Let K be a compact set in Cn, 0(z) extends to C+i (see (6)) and 0 < 51 < 52. If K is (5i, 0)-regular at z0 e K, then K is (52, 0)-regular at z0.

5. The property of (S, ^-regularity

Further properties of 5-extremal function are associated with pluri-thin sets.

Definition 6. Let E C Cn and let E' be its limit point set. Then E is said to be pluri-thin at z0 if either z0 e E' or z0 e E' but there exists a neighbourhood U of z0 and a function u(z) e psh(U) such that

lim u(z) < u(z0).

z

z£E\{z0}

So, if the set E is not thin at the point z0, then for any plurisubharmonic function u(z) in the neighborhood of z0

lim u(z) = lim u(z) = u(z ).

Z^rZ? Z^zP

z£E\{z0} zEE

Proposition 4 ([16]). If E c Cn is pluri-thin at a limit point z0 of E, then there exists a plurisubharmonic function u e L+ such that

limo u(z) = —to < u(z ).

zeE\{z0}

Theorem 5.1. If z0 is a pluri-thin point of K, then z0 is locally (S,')-irregular point of K. Here the function ' e L(K) and S e A.

Proof. Let K be pluri-thin at the point z0 e K. Then, according to Proposition 4, there exists a function u(z) e Lg such that

limo u(z) = —to < u(z0).

z£E\lz0}

Without loss of generality, we can assume u(z0) > 0 and find a ball B(z0,r) such that u(z) < inf '(z) — '(z0) for z e K n B \ {z0},

zEK u(z0) > 0.

Put w(z) = u(z) + '(z0). It is easy to see that w(z) e Lg(',K n B \ {z0}), because for z e K n B \{z0}

w(z) = u(z) + '(z0) < inf '(z) — '(z0) + '(z0) = inf '(z) < '(z).

z^K z^K

Consequently,

w(z) < Vs*(z,K n B \ {z0},') = Vg(z, K n B,'), Vz e Cn.

From here

w(z0) < Vg(z°, K n B,').

On the other hand

w(z0) = u(z0) + '(z0) > '(z0).

Therefore

'(z0) < w(z0) < Vs*(z0, K n B,'). Hence, the point z0 is a locally (S, ') irregular point of the compact set K. □

Note that if n > 1, the necessary condition of Theorem 5.1, generally speaking, is not true. Example 4. Let (S,')= (1,0) and K = {(zi,z2) e C2 : \z\ < 1} U {(zi,z2) e C2 : z2 =0, \zi \ < 2}. The compact set K is a union of the unit ball in C2 and a pluripolar set. We have

'ln+ ln+

V (z,K ) w

z\ for z2 = 2

z\

2

for z2 =0

and

V*(z, K) = ln+ \z\. A point (2,0) G K is an irregular point, but it is not pluri-thin.

References

[1] P.Lelong, Plurisubharmonic functions and positive differential forms, Gordon and Breach, New York, 1968.

[2] V.P.Zakharyuta, Bernstein's theorem on optimal polynomial approximations to holomorphic and harmonic functions of several variables, All-Union Conf. Theory of functions, Khar'kov, 1971, 80-81.

[3] V.P.Zakharyuta, Extremal plurisubharmonic functions, orthogonal polynomials, and the Bernstein-Walsh theorem for analytic functions of several complex variables, Ann. Polon. Math, 33(1976), 137-148.

[4] J.Siciak, Extremal plurisubharmonic functions in Cn, Proceedings of the First Finnish -Polish Summer School in complex Analysis at Podlesice, Lods: University of Lods, 1977, 115-152.

[5] A.Sadullaev, Plurisubharmonic measures and capacities on complex manifolds, Russian Math. Surveys, 36(1981), no. 4, 53-105.

[6] A.Sadullaev, Pluripotential theory. Applications, Palmarium Academic Publishing, 2012 (in Russian)

[7] A.Sadullaev, Pluriregular compacts in Pn, Contemporary mathematics. AMS, 662(2016), 145-156.

[8] H.Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., 36(1934), 63-89.

[9] A.Zeriahi, Fonction de Green pluricomplexe a pole a l'infini sur un espace de Stein parabolique, Math. Scand, 69(1991), 89-126.

10] M.A.Alan, Weighted regularity and two problems of Sadullaev on weighted regularity, Complex Analysis and its Synergies. Springer, 8(2019), no. 5.

DOI: 10.1007/s40627-019-0026-4

11] M.A.Alan, N. Gokhan Gogiis, Supports of weighted equilibrium measures: complete characterization, Potential Analysis, 39(2013), no. 4, 411-415 (SCI).

12] T.Bloom, N.Levenberg, Weighted pluripotential theory in CN, Amer. J. Math., 125(2003), no. 1, 57-103.

13] T.Bloom, Weighted polynomials and weighted pluripotential theory, Trans. Amer. Math. Soc., 361(2009), no. 4, 2163-2179.

14] M. Klimek, Pluripotential theory, Oxford University Press, New York, 1991.

15] N.X.Narzillaev, About 0-regular points, Tashkent. Acta NUUz, 2(2017), no. 2, 173-175 (in Russian).

[16] E.Bedford, B.A.Taylor, A new capacity for plurisubharmonic functions, Acta Mathematica, 149(1982), 1-41.

Дельта-экстремальная функция в пространстве Сп

Нурбек Х. Нарзиллаев

Национальный университет Узбекистана Ташкент, Узбекистан

Аннотация. В этой статье мы изучаем (5, ^-экстремальную функцию Грина У$(г,К,ф), которая определяется при помощи класса = {и(г) £ рвН(С") : и(г) < Си + 51п+ \г\, г € С"}, 5 > 0. Покажем, что понятие регулярности точек для разных 5 не совпадают. Тем не менее мы доказываем, что если компакт К С С" регулярен, то 5-экстремальная функция Грина непрерывна во всем пространстве С".

Ключевые слова: плюрисубгармонические функции, экстремальная функция Грина, функция Грина с весом, 5-экстремальная функция.