P-measure in the class of m−wsh functions Текст научной статьи по специальности «Математика»

УДК 517.55+517.947.42

P-Measure in the Class of m—wsh Functions

Bakhrom I. Abdullaev*

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

Uzbekistan

Received 04.10.2013, received in revised form 16.11.2013, accepted 09.01.2014 In this work we study the P-measure and P-capacity in the class of m-wsh functions and prove a number of their properties.

Keywords: m — wsh function, P-measure, P-capacity, mw-regular point.

Introduction

The classical potential theory (see [1, 2]) works with classes of harmonic and subharmonic functions and involves such concepts like the condenser capacity, harmonic measures of the sets, polar sets and others. The pluripotential theory, as is known, deals with the class of psh functions and the Monge-Ampere operator (ddcu)n = 0 (see [3, 4]), where as usual

- d — d

d = d + d, dc =-.

4i

In the recent work [5] the author has studied the class of m — wsh functions, introduced the concept of mw-polarity of sets and proved several of their properties. In this paper we study the p-measure and p-capacity in the class of m — wsh functions. In section 1 we briefly give the definition of m — wsh functions and some results, which we use below. In section 2 we give the definition of p-measure and we prove some of its properties. Section 3 is dedicated to the p-capacity and its properties.

We note that m — sh and m — wsh functions are related to the Hessians of function u (see [3, 6]). They can be used in different problems of multidimentional complex analysis. One of such application is shown in the work [6] (see also [7, 8]) where the characteristic functions of Nevalinna of higher order are estimated.

1. m—wsh functions

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

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

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

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

2) the current ddcu A jn m defined on C— smooth and finite (m — 1,m — 1) forms w as

ddcu A jn-m (w) = J u[3n-m A ddcw > 0, Vw g F(m-l,m-l),

is positive, i.e. ddcu A jn-m ^ 0. We note that a differential form

w = (^J (dl\ A dii) A ... a (dlp A dip)

is called the main positive form of bidegree (p,p), 0 < p < n, where lj = aj1 zi + ... + ajnzn are linear functions in Cn, j = 1,2,...,p.

A linear combination of such forms wq

N

w(p,p) = £ fq (z)wq, fq (z) G L}oc(D) , fq (z) > 0, q=i

is called a strongly positive differential form of bidegree (p,p) in D c Cn.

Thus, a positive differential form of bidegree (0,0) or bidegree (n, n) give us a positive scalar function w(0'0) = f (z) ^ 0 or

w(n'n) ^2) f (z)dzi A dzi A ... A dzn A dzri = f (z)dV, f (z) > 0,

where dV is the Lebesgue volume element in the space Cn ~ R2n.

A differential form w(p'p) G F(p'p) of bidegree (p,p) is called weakly positive, if w(p'p) A a ^ 0 is a positive form of bidegree (n, n) for any strongly positive form a G F (n-p,n-p).

A strongly

positive form is, at the same time, weakly positive, because the exterior product of two strongly positive forms is positive.

Definition 2. A linear continuous functional T(w) on the space of differential forms

F(p,p) = F(p,p)(D) = {w G F(p,p)(D) n C™(D) : suppw cc D}

is called a current of bidegree (n — p,n — p) = (q, q).

A 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 the weak positivity of a current is equivalent to the strong positivity.

It is known that positive currents are the measure type currents, that is differential forms with coefficients that are Borel measures. For more details about the theory of currents see [9, 10]. An important example of the currents of bidegree (p,p) in the potential theory is the currents ddcu A j3p-1, 1 < p < n, defined as

ddcu A 3p-1 (w) = j ujp-1 A ddcw , w g F(p'p)(D), (1)

where u G Lloc(D) is a fixed function and 3 = ddclz\2 is the volume form in Cn.

It is not difficult to prove that a current of the form ddcu A j p-1 is strongly positive if and only if it is weakly positive.

We consider the following properties of the m — wsh functions that we need further.

1) A linear combination of m — wsh functions with non-negative coefficients is an m — wsh function, that is

uj(z) £ m — wsh(D), a,j £ R+ (j = 1, 2,..., N) ^ aiui(z) + a2u2(z) + ... + a^un(z) £ m — wsh(D);

2) The limit of a monotone decreasing sequence of m—wsh functions is an m — wsh function, that is

uj(z) £ m — wsh(D), uj (z) ^ uj+i (z), (j = 1, 2, ...) ^ lim uj (z) £ m — wsh(D);

j — TO

3) A uniformly convergent sequence of m—wsh functions converges to an m—wsh function, that is if uj(z) £ m—wsh(D), j = 1, 2,..., and if uj(z) ^ u(z), then u(z) £ m—wsh(D);

4) (The maximum principle) Let u(z) £ m—wsh(D) and at some point z° £ D u(z) attains its maximum, that is

u(z0) = sup u(z), (2)

zED

then u(z) = const.

5) If u (z) £ m — wsh (D), then the convolution uj (z) = u * Ki/j (z — w) also belongs to m — wsh (D) and uj (z) j u (z) as j ^ to. Here Ki/j(z) = jnK (jz), and K is the standard infinitely smooth kernel with the support suppK c B (0,1) and averaging

i K (z) dV = i K (z) dV = 1.

JRn ^B(0,i)

Theorem 1. An upper semicontinuous function u given in D c Cn is m — wsh if and only if for any (n — m + 1)-dimensional complex plane Yl c Cn the restriction is a subharmonic function in n p| D.

2. P-measure

Let E c D be an arbitrary set of D c Cn, 1 < m < n. For simplicity, we assume that D is a strictly mw-convex domain, that is D = {p(z) < 0}, where p(z) is a strictly m —wsh function in some neighborhood of G d D. Recall that a twice-smooth function p (z) £ C2 (D) is called strictly m—wsh in the point z° £ D, if the operator ddcu a is strictly positive, i.e. for some positive number 5 > 0, ddcu a pn-m ^ 5^n-m+i in a neighborhood of z°. It is called strictly m—wsh in the domain D if this is true at each point z° £ D. Consider the class of functions

U(E, D) = {u £ m — wsh(D) : u |D < 0, u |E < —1}

and put

w(w, E, D) = sup{u(w) : u £ u(E, D)}. Definition 3. The regularization u>* (z, E, D) = lim w (w, E, D) is called p-measure (m — wsh

w—>z

measure) of the set E with respect to D.

Here there are some simple properties of p-measure.

1) (monotony) If E1 c E2, then w*(z,EbD) > w*(z,E2,D). If E c D1 c D2, then w*(z,E,D^ > w*(z,E,D2);

2) w*(z, U,D) g u(U,D) for open sets U c D and therefore w*(z,U,D) = w(z,U,D). In fact, it is easy to prove that w*(z, E, D) = —1 if z is an interior point of the set E, z g E0. Hence w*(z, U, D) = —1 for any z g U and w*(z, U, D) g u(U, D). It follows that w*(z, U, D) < w(z, U, D) and, therefore w*(z, U, D) = w(z, U, D).

3) If U c D is an open set and U = |J°=1 Kj, where Kj c K , then w*(z, Kj, D) j w(z, U, D)

j j+i

(It follows easily from property 2).

4) If E c D is an arbitrary subset, then there is a sequence of open sets Uj d E, Uj d Uj+1 (j = 1, 2,...), such that w*(z, E, D) = [lim w(z, Uj, D)]*. Indeed, by Choquet's Lemma

j — TO

(see [1]) there exists a countable family u' c u such that < sup u (z) > = < sup u (zU =

[«ew' ) Ueu J

w* (z, E, D).

Hence if u = {u1,u2,...} and Vj (z) = max {u1 (z),...,uj (z)}, then Vj (z) j v(z) and v* (z) = w* (z, E, D). Now, if we put Uj = jVj < —1 + 11, then Uj is open and E c Uj c D.

Therefore, w* (z, E, D) > w* (z, U, D), j = 1,2,.... On the other side v, — 1 g u (U, D) and

Vj — 1 < w (z, Uj, D). Consequently, lim Vj (z) < lim w (z, Uj, D) and v* (z) = w* (z, E, D) < j j—j—

*

. □

lim w (z, Uj, D)

5) The p-measure w*(z, E, D) is nowhere equal to zero or identically zero. w*(z, E, D) = 0 if and only if E is mw-polar in D (it is proved analogously to the corresponding property of the potential theory)

6) (theorem of two constants) If in the domain D c Cn a function u(z) g m — wsh and u |D < R, u |E < r, (E c D), then for all z g D we have the inequality

u(z) < R(1 + w*(z,E,D)) — rw*(z, E, D).

This follows from the fact that the function

u (z) — R

R —

gw (e,d) .

7) If the set E compactly lies in a strictly mw- convex domain D = {p (z) < 0} , E cc D, then the p-measure w*(z, E, D) m — wsh continues to G, G d D.

Indeed, from the condition E cc D it follows that there exists a constant M > 0 such that M • p (z) < —1, z g E. Hence, w*(z, E,D) > M • p (z), z g D. Therefore, the function

w (z) = I w*(z,E,D) , ifz g d 1 ; \ M • p (z) , ifz g D

is m—wsh in G which gives continuation of the p-measure in G.

Definition 4. z0 g K is called an mw-regular point of the compact K, if w*(z0,K, D) = —1. K c D is called an mw-regular compact if every its point z0 g K is mw-regular.

Note that the regular compacts are always mw-regular. It is well known that if the boundary of a bounded open set G consists of a finite number of twice smooth surfaces, G is a regular compact.

This implies that for any compact K C U C D, where U is an open set, there is always an mw-regular compact E : K C E C U C D.

The next theorem is very important in the study of m — wsh functions.

Theorem 2. If K C D is an mw-regular compact, then the P-measure v*(z, K, D) = v(z, K, D) and is a continuous function in D.

(It can be proved similarly to the continuity of P-measures in the pluripotential theory).

3. P-capacity

Let E C D and v*(z, E, D) be its p-measure. The value

P(E,D) = —[ v*(z,E,D)dV

J D

is called the P-capacity of the set E with respect to D.

Thus, the p-capacity expresses the capacitive value of the pair (E, D). Such a pair is usually called a condenser in Cn. For m = n the P-capacity was introduced by Sadullayev (see [8]). It has been used in the estimates of volumes of analytic sets, in the descriptions of the family of defective divisor of a holomorphic mapping, etc. (see [8]). The p-capacity has the following properties:

1) P(E,D) = — fD v(z, E, D)dV (this follows from the fact that the set {v(z,E,D) < v*(z, E, D)} is a polar set. Consequently, it has zero Lebesgue measure);

2) P(E, D) > 0 and P(E, D) = 0 if and only if E is an mw-polar set in D (the proof follows from the definition of the P-capacity and the relevant property of P-measure).

Measurability of the p-capacity P(E, D) is based on the following theorem.

Theorem 3. The value P(E,D) is an increasing and countable sub-additive function of the set: P(Ei,D) < P (E2,D) for Ei C E2 and

(OO \ OO

U EjP (Ej, D). (3)

j=i J j=i

Moreover, P(E, D) is right-continuous, that is for any set E C D and for any e > 0 there exists an open set U d E such that P (U, D) —P (E, D) < e

Proof. Monotony of P(E, D) clearly follows from the monotonicity of the p-measure. The proof of (3) follows from a similar inequality

OO

—u\z,[j Ej,D I < v (z,Ej, D) j=i ) j=i

for p-measure: for any set of Uj (z) £ U (Ej, D) the sum Y1 uj (z) is an m — wsh function in

j=i

the wide sense (that is, it can also be equal to —to). Moreover, J2 uj (z) & U [ UO=i Ej, D ) and

j=i v '

hence uj (z) < v Uj=1 Ej, Dj.

Now

{O | O O

]T Uj (z) : Uj (z) e U (Ej, D)\=Y^ sup {uj (z) : Uj (z) e U (Ej,D)} = £ v (z, Ej,D),

j=i ) j=i j=i

and

O I O \

Y^v (z, Ej, D) < vlz,[jEj.

j=i \ j=i J

Integrating this inequality and using Levi's theorem, we get

J ш (j Ei, D j dv < - J2 J ш (z, Ei' D) dV,

TO

p UEj ,D) P (E,D) ■

\j=i J j=i

To show the right-continuity of the set-function P(E, D), we fix the set E с D and according to property of p-measure we construct open sets Uj D E, Uj D Uj+\\ ш* (z, E, D).

As ш (z, Uj, D) is increasing, then again by Levi's theorem

lim ш (z, Uj, D)

j — TO

lim p (Uj,D) = - lim [ ш (z, Uj, D) dV = - i lim ш (z, Uj,D) =

j—>TO j—>TO J J j—

lim ш (z, Uj, D)

j — TO

dV = P (E, D).

Hence, for any e > 0 there is such j0, when j > j0 we have the inequality P (Uj,D) — P (E, D) < e. The theorem is proved. □

Corollary 1. For any decreasing sequence of compacts Ki D K2 D ... we have the equality

P I I J Kj ,D | = lim P (Kj, D) j )

If Gi C G2 C ...are the sequences of open sets and G = |JO=i Gj then

P (G,D)=lim P (Gj, D). (4)

Corollary 2. The set function P(E,D) has all the properties of Choquet measurability (see [1, 2]) and therefore any Borel sets are measurable with respect to the capacity P.

Thus, if E C D is a Borel set, then its interior and exterior capacities coincide: *P (E, D) = P* (E, D) = P (E, D), where P* (E, D) = inf {P (U,D) : U D E is open} and

*P (E, D) = sup {P (K,D) : K C E is compact} .

Remark. In the classical potential theory and in the pluripotential theory (4) is proved for any arbitrary sequence of sets Ei C E2 C .... It is based on the following problem, which solution is unknown to us: let {uj (z)} be a locally uniformly bounded, monotonically increasing sequence of m — wsh functions and u (z) = lim Uj (z). Then will there be the set {u (z) < u* (z)} mw-polar

in D?

TO

*

References

[1] M.Brelot, Foundations of classical potential theory, Wiley, NY, 1964.

[2] N.S.Landkof, Foundations of modern potential theory, Nauka, Moscow, 1966 (in Russian).

[3] A.Sadullaev, Plurisubharmonic measures and capacities on complex manifolds, Advances math. Sciences, 36(1981), no. 4, 53-105.

[4] E.Bedford, B.A.Taylor, A new capacity for plurisubharmonic functions, Acta Math., 149 (1982), no. 1-2, 1-40.

[5] B.I.Abdullaev, Subharmonic functions on complex hypersupfaces of6Cn, Journal of Siberian Federal University. Mathematics & Physics, 6(2013), no. 4, 409-416.

[6] B.I.Abdullaev, F.K.Ataev, Nevanlinna's characteristic functions with complex Hessian potential (preprint).

[7] B.V.Shabat, Distributions of values of holomorphic mappings, Translations of Mathematical Monographs, v. 61, Providence, RI, 1985.

[8] A.Sadullaev, Deficient divisors in the Valiron sense, Mat. Sb., 108(1979), no. 4, 567-580.

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

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

[11] Z.Blocki, Weak solutions to the complex Hessian equation, Ann. Inst. Fourier (Grenoble), 55(2005), no. 5, 1735-1756.

P-мера в классе m—wsh функций

Бахром И. Абдуллаев

В данной 'работе изучена P-мера и P-емкость в классе m — wsh функций, также доказаны их

некоторые свойства.

Ключевые слова: m — wsh функции, P-мера, mw-регулярная точка, P-емкость.