Subharmonic functions on complex hyperplanes of c n Текст научной статьи по специальности «Математика»

УДК 517.55

Subharmonic Functions on Complex Hyperplanes of Cn

Bakhrom I. Abdullaev*

Urgench State University, H.Olimjan, 14, Urgench, 220100, Uzbekistan

Received 06.06.2013, received in revised form 02.07.2013, accepted 15.08.2013

In this paper is considered a class of m — wsh functions defined with relation ddcu A (ddc|z|2)n-m ^ 0, and is studied some properties of polar sets for this class.

Keywords: m — wsh function, mw-polar set, mw-convex domain, mw-regular domain.

Introduction

Subharmonic (sh) and plurisubharmonic (psh) functions play the main role in theory of functions of several real and complex variables. In the space Cn « R2n they defining by the conditions

ddcu A (ddc |z|2)n-1 ^ 0

or

ddcu ^ 0,

— d — d

respectively. Here, as usual d = d + d, dc = ———.

4*

In this paper we consider the class of m-weak subharmonic (m — wsh) functions, defined by relation

ddcu A (ddc|z|2)n-m > 0. (1)

As we see below this class wider than the class of psh functions, but strongly contains in the class of sh functions. Moreover, in case, m =1 the class of 1 — wsh functions coincide with class of sh functions and in case m = n the class of n—wsh functions coincide with class of psh functions.

In studying the class of m—wsh functions we essentially use the elementary theory of differential forms and currents, also methods of pluripotential theory. In general case, when u isn’t twice differentiable, the relation (1) is interpretated in the sence of currents. Therefore in section 1 we shortly give fundamental conceptions from the theory of currents. In section 2 we give general definition of the m — wsh functions and some their simple properties. Section 3 devoted to the mw-polar set and its characteristics.

1. Positive defined differential forms and currents

As usual, the space of differential forms of bidegree (p, p) in a domain D C Cn is denote by F(p,p) = F(p,p)(D). The differential form in view

(dli A dl]_) A ... A (dlp A dip)

*[email protected] © Siberian Federal University. All rights reserved

is called main positive form of bidegree (p, p), 0 < p < n, where lj = a^ zi + ... + ajn zn are linear functions in the space Cn, j = 1,2, ...,p. Linear combination of such form wq

N

^(p,p) = £ fq (z)Wg, fq (z) G C(D), fq (z) > 0,

q=i

is called strongly positive differential form of bidegree (p,p) in the domain D c Cn. Thus, positive differential form of bidegree (0, 0) or bidegree (n, n) give to us positive scalar function

w(0,0) = f (z) > 0 or

n

i

w(n’n) = f (z)dzi A dzi A ... A dzn A dzn = f (z)dV, f (z) > 0,

where dV — Lebesgue’s element of volume in the space Cn ~ R2n.

The differential forms w(p,p) G F(p,p) of bidegree (p,p) is called weakly positive if w(p,p) A a is positive form of bidegree (n, n) for any strongly positive form a G F(n-p>n-p). Strongly positive form is at the same time weakly positive, because exterior product of two strongly positive form are positive.

In the cases p = 0,1, n—1, n weakly and strongly positive are coincide. But, in cases 1 < p <

n - 1 not every weakly positive differential form is strongly positive.

Definition 1. Linear continuous functional T(w) in the space of main differential form F(p’p) = F(p’p)(D) = {w G F(p’p)(D) n CTO(D) : supp w cc D} is called current of bidegree (n — p, n — p) = (q, q)

The current T is called strongly (weakly) positive, if T(w) > 0 for any weakly (strongly) positive form w G F(p’p). It is clear, that for q = 0,1, n—1, n weakly positivity of currents also coincide with strongly positivity.

It is known, that positive currents are currents of measure type, i.e. differential forms, coefficients which are Borel’s measures. More about the theory of currents see [1-4].

An impotent example of current of bidegree (p, p) in the pluripotential theory is current ddcu A (ddc|z|2)p-i, 1 ^ p ^ n, defined as

ddcu A (ddc |z|2)P~ ^w)^ u(ddc |z|2)P_ *A ddcw,w G F(n-p>n-p)(D), (2)

where u G L;1oc(D) are fixed functions. It is easy to proof that the current ddcu A (ddc|z|2)p-i is

strongly positive if and only if, when it is weakly positive.

2. m-wsh functions

Definition 2. A function u(z) G L^^D), given in a domain D c Cn is called m — wsh function (subharmonic function on (n — m + 1) -dimensional complex surfaces, 1 ^ m ^ n) in D if:

1) it is upper semicontinuous in D, i.e.

lim u(z) = lim sup u(z) ^ u(z0);

B(z0,e)

2) the current ddcu A (ddc|z|2|)n-m ^ 0 in D, i.e.

ddcu A (ddc|z|2)n m (w) = J u (ddc|z|2)n m A ddcw > 0, Vw G F(m-i-m-i), w > 0.

The class of such functions is denoted by m—wsh(D). For convenience, the function u = —to also included into the m — wsh(D) class. A letter "w" (weak) in denotation of class is put in order to differ this class from the known class of m — sh functions. m — wsh function in the domain D c Cn at the same time is subharmonic in the D c R2n. Therefore, all properties of subharmonic functions is true for m —wsh functions.

We provide a following properties of m —wsh function, which we will use further.

1) Linear combination of m—wsh functions with nonnegative coefficients are m—wsh functions,

i.e.

uj(z) G m — wsh(D), aj G R+ (j = 1, 2,..., N) ^

aiui(z) + a2u2(z) + ... + aNun(z) G m—wsh(D).

2) A limit of monotonically decreasing sequences of m — wsh functions is m — wsh function,

i.e.

uj(z) G m — wsh(D), uj(z) > uj+i(z), (j = 1, 2,...) ^

lim uj(z) G m —wsh(D).

j^TO

3) Uniformly convergence of sequence of m — wsh functions is converge to m — wsh function, i.e. if uj(z) G m—wsh(D), (j = 1, 2, ...), uj(z) ^ u(z), then u(z) G m—wsh(D).

4) (maximum principle). Let a function u(z) G m — wsh(D) and in some point z0 G D it reaches its maximum, i.e.

u(z0) = sup u(z). (3)

Then u(z) = const.

5) If u(z) G m—wsh(D), then a convolution uj(z) = u*Ki/j(z—w) also belongs to m—wsh(D), and uj (z) j u(z) at j ^ to.

Here Ki/j(x) = jnK (jx) and K is standard infinity differentiable kernel, with carrier suppK c B(0,1) and

/ K (x)dx = K (x)dx = 1.

-'B(0,i)

The proof of these properties implies from analogous properties of subharmonic functions on the plane and we down them (in details see [5]).

A following theorem gives us geometric character of m— wsh functions.

Theorem 1. Upper semi continuous function u, given in the domain D c Cn, is m — wsh if and only if for any (n — m + 1) -dimensional complex surface n c Cn restriction

u|n G sh (n n D). (4)

Proof. Necessity. Let u G m — wsh(D). According to property 5 we approximate u, with infinity differentiable functions uj j u, uj G m — wsh(D) n CTO(D). We fix a complex plane n c Cn, dimC n = n — m + 1, and we take an orthonormal basis ..., £n-m+i on n. Then (ddc|z|2)n-m |n = (ddc|£|2)n-m and consequently, ddcuj A

(ddc|z|2)n-m |n = ddcuj |n A (ddc|£|2)n-m . Since, ddcuj A (ddc|z|2)n-m is positive differential form of bidegree (n — m + 1, n — m +1), then the restriction ddcuj A (ddc|z|2)n-m |n ^ 0. Hence ddcuj |n A (ddc|£|2)n-m > 0 and it means, that uj |n G sh(n n D). Since, uj |n j u |n at j ^ to, then u |n G sh(D).

Sufficiency. First we formulate a number of properties of upper semi continuous function u(z), satisfying the condition (4), by them we will proof of sufficiency of theorem.

1) Finite sum aiui + ... + akuk with positive coefficients ai,...,ak > 0 will satisfy the condition (4), if and only if ui, ...,uk satisfy the condition (4).

2) Decreasing sequence or uniformly convergence sequence of functions {uj}, satisfying the condition (4) converges to function of type (4).

3) The function u, satisfying the condition (4) either u = —to, or locally summable function,

Le. u e LL(d).

Indeed, since u is upper semicontinuous, then it locally bounded from above. Therefore, without lost of generality we may assume, that u < 0 in D. Let in some point z0 = 0 the function u(0) = —to. Then for any fixed surface n 9 0, dimn = n - m +1, the restriction u |n is subharmonic in D n n. Consequently,

u (0) ^ V----------1rn-m+^ u |n dV|n, (5)

Vn-m+1r J B(0,r)nn

where B(0, r) = {||z|| < r} is a ball, dV|n is an element of volume on n and Vn-m+1 is a volume of unit ball in n ~ Rn-m+1. Hence, for any surface n 9 0, dimn = n-m +1, the restriction u |n has uniformly bounded integrals on n n B(0, r). By the Fubini theorem and according to (5) it

follows that, —to < u(z)dz < 0. It means, that u locally integrable in a neighbourhood

J B(0,r)

of origin and it follows that the function u integrable on any Ball B(z0, r), z0 e D, r > 0. Remark 1. Here we apply the Fubini theorem on collection of complex surfaces passing through origin. As it is known they generate Grassman’s manifold Mn,n-m+1. But to prove locally integrability of u we can apply the theorem of Fubini for all complex surfaces n passing through some fixed surface L 9 0, dim L = n - m. The set of such n will generate a projective space Pm-1, and to proof u e L^D) we can use a following convenient formula of Fubini

f u(z) dv = f wm-1 f u|n(z) dV|n, (6)

J B(0,r) JnePm-1 J B(0,r)nn

where w is standard form of Fubini-Shtudi of projective space.

4) If u satisfy the condition (4), then the convolution uj (z) = u * K1/j (z - w) also satisfy this condition and uj(z) j u(z) at j ^ to.

It follows from obviously relation

u * K1/j (z - w) = jn u(w)K (j(z - w))dw = ^(z +—(w)dw. (7)

J Rn J Rn V j /

Here, the first integral represents infinity differentiable function, second integral satisfies the condution (4). Convergence of uj (z) j u(z) follows from (6).

Now we can complete the proof of theoreml. According to property 4) we construct approximation uj(z) j u(z). Since, uj e and uj |n are subharmonic on any complex surface n, dimCn = n - m +1, then the restriction ddcuj A (ddc|z|2)n-m |n ^ 0. It means, that the differential form ddcuj A (ddc|z|2)n-m ^ 0. From convergence of uj (z) j u(z) follows ddcu A (ddc|z|2)n-m ^ 0 in the sence of currents, and consequently, u e m-wsh(D). The proof of theoreml is complete. □

3. mw-polar sets

The polar and pluripolar sets are key notions of the potential theory (see [3,6]). Therefore, it is important the study of the mw-polar sets for the class of msh-functions.

Definition 3. By analogue polar sets, a set E C D C Cn is called mw-polar in D, if there exist a function u(z) e m-wsh(D), u(z)= - to, such that u |E = - to.

From inclusion m - wsh(D) C sh(D) it is follows, that each mw-polar set is polar. In partiqular, the Hausdorf measure H2n-2+e (E) =0 Ve > 0, and consequently, Lebesgue measure of mw-polar set E also is zero.

From embedding psh(D) C m - wsh(D) follows, that every pluripolar set is mw-polar. We provide a nontrivial example of mw-polar set in the space C3.

Example 1. We consider a function

u = ln[(z1 + Z1)2 + (z2 + z2)2 + (z3 + z3)2] = ln |z + z|2 = ln (^2 + x2 + £3) + ln4,

where zj = £j + iyj, j = 1, 2, 3.

It is clear u is not 3 - wsh in D, i.e. it is not psh in D. It is not difficult to prove that it is subharmonic, i.e. Au ^ 0. We show that it is 2 - wsh function in C3. Thereby we have, that real 3-dimentional surface R3(x) = {z e C3 : Imz = 0} is 2w-polar in C3. Taking direct calculation.

w = (ddcu) A ddc |z|2 = —

d2?

dz1dz1

dz1 A dz1 +

d2?

dz1dz2

dz1 A dz2 +

d2u d2u d2u d2u

+ ~—dz1 A dz3 + ——dz2 A dz1 + ——dz2 A dz2 + -r—dz2 A dz3+

dz1dz3

dz2dz1

dz2^z2

dz2^z3

d2u d2u d2u

+ ——dz3 A dz1 + ——dz3 A dz2 + ^dz3 A dz3

dz3dz1

dz3dz2

dz3

A

—1

A^(dz1 A dz1 + dz2 A dz2 + dz3 A dz3) = -4

d2u d2u .

+ ——) dz1 A dz1 A dz2 A dz2 +

+

+

\dz1dz 1 dz3dz3 d2u

dz1 A dz1 A dz3 A dz3 +

dz1dz 1 dz2dz2

d2u d2u

+

dz2dz2 dz3dz3

dz2 A dz2 A dz3 A dz3+

dz2^z3

d2u

d2u d2u

dz1 A dz1 A dz2 A dz3 + ——7— dz1 A dz1 A dz3 A dz2 + -r—dz1 A dz A dz2 A dz2+

dz3dz 2

dz1dz3

d2u d2u

o dz3 A dz1 A dz2 A dz2 + ^—^rz- dz1 A dz2 A dz3 A dz3 + ——7— dz2 A dz1 A dz3 A dz

dz3dz 1 dz1 d^2 dz2dz 1

Thus, for any form v = -dl A dl of bidegree (1,1) where

2

—

from v = — dl A 2

2

= «1dz1 + a2dz2 + «3dz3

—2

2 (J«1| dz1 A dz1+ a1a2dz1 A dz2 + «1 ^dz! A dz3 + a2«1dz2 A dz1 + + |«212 dz2 A dz2 + a2«3dz2 A dz3 + a3«1dz3 A dz1 + a3«2dz3 A dz2 + |«312 dz3 A dz^, we get

v A w =----

8

M2

d2u d2u

+

dz2dz2 dz3dz3

dz1 A dz 1 A dz2 A dz 2 A dz3 A dz 3 +

d 2u d2u

+«1«2^—dz1 A dz2 A dz2 A dz1 A dz3 A dz3 + 0103 ———- dz1 A dz3 A dz3 A dz1 A dz2 A dz2 +

dz2dz 1 dz3dz 1

d 2u

+a2 «1 o—dz2 Adz 1 Adz1 Adz2 Adz3 Adz3+1 «212

+a2«3

+a3«2

dz1dz2

d 2u dz3dz2 d 2u dz2dz3

d2u d2u

+ -

dz1dz1 dz3dz3

dz1Adz 1Adz2Adz 2Adz3Adz3+

dz2 A dz3 A dz1 A dz 1 A dz3 A dz2 + 0301

d2u

dz1dz3

dz3 A dz 1 A dz1 A dz2 A dz2 A dz3+

dz3 Adz2Adz1 Adz1Adz2Adz3+|a3|2

d2u d2u

- + -

dz1dz 1 dz2dz2

dz1 Adz1 Adz2 Adz2 Adz3Adz3 =

2 ( d2u d2u \ 2 ( d2u d2u \ , ,2 f d2u d2u

|fl11 a. a = + a. M a. a = + a. a^ +

\dz2dz2 dz3dz3) \dz1dz1 dz3dz3 J \dz1dz 1 dz2dz2

d2u _ d2u _ d2u _ d2u _ d2u _ d2u

— «1«2 ~e\--^ _ — «1 «3 "tt-_ _ — «2 «1 ~e\-^ _ — «2«3 Ta--^ _ — «3«1 ---- _ — «3«2------

4

dz2dz1 dz3dz1 dz1dz2 dz3dz2 dz1dz3 dz2dz3

— — — — — —

x — dz1 A dz1 A — dz2 A dz2 A — dz3 A dz3 = a(z) — dz1 A dz1 A — dz2 A dz A — dz3 A dz3,

2 2 2 2 2 2

where

t \ | |2 2 |z + z |2 — 4(z2 + z2)2 , 2 |z + z |2 — 4(z3 + z3)2 ^ ,

a(z) = |fl1 M ------------------------------------------ +-j—7^4----- +

V |z +z | |z +z | )

2 (2 |z + z | — 4(z1 + z1)2 2 |z + z | — 4(z3 + z3)2^\

+ ^------------------i“----------- +---------------------j—7^4- +

|z + z | |z + z |

2 (2 |z + z | — 4(z1 + z1)2 2 |z + z | — 4(z2 + z2)2^\

+ ^ ------------1 j774--------+---------------------[—7^4- +

|z + z | |z + z |

_ 4(z1 + z1)(z2 + z2) _ 4(z1 + z1)(z3 + ^3)

+a1«2---------- -----4-------------------+ a1«3------- --4------------------+

|z + z|4 |z + z|4

_ 4(z1 + z1)(z2 + ^2) _ 4(z2 + z2)(z3 + ^3)

+a2«1---------- -----4-------------------+ «2«3-;--------4------------------+

|z + z|4 |z + z|4

_ 4(z1 + z1)(z3 + £3) _ 4(z2 + z2)(z3 + z3)

+«3«1----------:-----4-------------------+ a3«2-:--------4----------------- =

|z + z|4 |z + z|4

= "--------4 f|a1| (z1 + z1)2 + |«2| (z2 + z2 )2 + |«3| (z3 + z3)2 + «1«2(z1 + z 1) (z2 + z2) +

|z + z |4

+a1«3 (z1 + z1)(z3 + 2:3) + a2«1(z2 + z2)(z1 + 21) + a2«3(z2 + z2)(z3 + 23) +

+a3«1(z3 + z3)(z1 + z1) + a3«2(z2 + z2)(z3 + z3)) =

42 = --4 |«1(z1 + z 1) + «2 (z2 + z2 ) + «3(z3 + z3)| ^ 0.

|z + z |4

Since, l-arbitrary linear function, then ddcu A ddc|z|2 ^ 0, in C3\R3(x) i.e. u is 2 — wsh function beyond of points R3(x). In points R3(x) function u |R3(x) = —to. Consequently, it will be automatically 2 — wsh in these sense.

Definition 4. A domain D C Cn is called mw-convex, if there exist p(z) e m — wsh(D) such that lim p(z) = +to, and it called mw-regular, if there exist p(z) e m—wsh(D) : p(z) < 0 such

that lim p(z) = 0.

z^dD

Next two theorems are analogue of corresponding theorems of classical and complex theory of potential (see for example [6,7]).

Theorem 2. Countable union of mw-polar sets is mw-polar, i.e. if Ej C D are mw-polar, then

CO

E = U Ej is also mw-polar.

j=1

Theorem 3. Let D C Cn be mw-convex domain and subset E C D such that for any compact subdomain G CC D the set E n G mw-polar in G. Then E is mw-polar in D. Moreover, if D-mw is regular, then there exist a function u(z) e m — wsh(D), u |D < 0, u= — to, but u |e = —to.

Proofs of these theorem close to eachother. Therefore we provide only proof of the Theorem 3. Since D is mw— convex domain, then a function p(z) = — ln p(z, dD) is m — wsh(D) and lim p(z) = +to. Hence, Dr = {z e dD : p(z) < r} CC D for any r > 0. We fix some point

z——dD

a e D and denote by Gj connected component of the set Drj, enclosed a point a. Then there exist a number rj > p(«) such that

O

Gj CC Gj+1, jGj = D. (8)

j=1

Since E n Gj+1 is mw-polar, then there exist a functions Vj(z) e m — wsh(Gj+2) such that

Vj = — to, but Vj |EnGj+2 = —to. As the set {vj = —to} has a Lebesgue measure zero, then the

OO

set U {vj = —to} also has a Lebesgue measure zero. Consequently, there is a point z0 e G1

j=1

such that Vj (z0) = —to for all j e N.

Putting Cj = max Vj(z), tj(z) = — • Vj^j and uj(z) = «j(p(z) — rj+1), where

ze<3,,-+i 2j Vj(z ) — Cj

«j > 0 so big, that u |Gj < —1. Then tj(z) |Gj-1 < 0 and uj |dcj+1 = 0. Therefore, it is not

difficult to proof, that

w (z) = / max{tj(z),uj(z)}, for z e (9)

wj(z) = \ uj(z), for z e Gj+1 (9)

is mw-subharmonic in D (j = 1, 2,...).

Then the sum w(z) = ^ wj(z) e m —wsh(D), and w (z^ = —1, w |E = —to. It follows that

j=1

E is mw-polar in D.

In the case, when D = {p(z) < 0} is mw-regular, i.e. p (z) e m — wsh (D) : p (z) < 0 and

lim p (z) = 0, as a set Dr = {z e dD : p(z) < —r} CC D, r > 0, and as a function uj we

z—dD

take uj (z) = «j [p(z) + rj+1]. Here the sequence rj j 0 such, that the connected component Gj of Drj satisfy the condition (8) and the «j a such, that u |Gj < —1. Further, we construe wj as

O

in (9) and we put w(z) = ^ wj(z). Then w will be at first negative m—wsh function in D and

j=1

secondly w |E = —to. □

The work was supported in part by the grant of fundamental researchs F4-FA-0-16928 of Khorezm Mamun Academy

References

[1] E.M.Chirka, Complex analytic sets, Moscow, Nauka, 1985 (in Russian).

[2] J.P.Demailly, Measures de Monge-Ampere et measures Plurisousharmoniques, Math. Z., 194(1987), 519-564.

[3] M.Klimek, Pluripotential Theory, Clarendon Press, Oxford-New York-Tokyo, 1991.

[4] P.Lelong, Fonctions plurisousharmoniques et formes differentielles positives, Dunod, Paris, Gordon & Breach, New York, 1968.

[5] W.K.Hayman, P.B.Kennedy, Subharmonic functions, Vol. 1, London Math. Society Monographs, no. 9, Academic Press, 1976.

[6] A.Sadullaev, Plurisubharmonic measure and capacity on complex manifolds, Russian Math. Surveys, 36(1981), 61-119.

[7] S.Kolodziej, The complex Monge-Ampere equation and pluripotential theory, Mem. Amer. Math. Soc., 178(2005), 64.

Субгармонические функции на комплексных гиперплоскостях Cn

Бахром И. Абдуллаев

В данной статье 'рассмотрен класс m — wsh функций, определяемых соотношением ddcuA A(ddc|z|2)n-m ^ 0, и изучены некоторые свойства полярных множеств из этого класса.

Ключевые слова: m — wsh функции, mw-полярное множество, mw-выпуклая область, mw-регулярная область.