REMOVABLE SINGULARITIES OF SEPARATELY HARMONIC FUNCTIONS Текст научной статьи по специальности «Математика»

DOI: 10.17516/1997-1397-2021-14-3-369-375 УДК 517.55

Removable Singularities of Separately Harmonic Functions

Sevdiyor A. Imomkulov*

Khorezm Regional Branch of the V. I. Romanovsky Mathematical Institute Academy of Sciences of the Republic of Uzbekistan

Urgench, Uzbekistan

Sultanbay M. Abdikadirov"

Karakalpak State University Nukus, Uzbekistan

Received 20.01.2021, received in revised form 09.02.2021, accepted 09.03.2021 Abstract. Removable singularities of separately harmonic functions are considered. More precisely, we prove harmonic continuation property of a separately harmonic function u(x,y) in D \ S to the domain D, when D С Rn(x) x Rm(y), n,m> 1 and S is a closed subset of the domain D with nowhere dense projections Si = {x € Rn : (x,y) € S} and S2 = {y € Rm : (x,y) € S}.

Keywords: separately harmonic function, pseudoconvex domain, Poisson integral, PP-measure.

Citation: S.A. Imomkulov, S.M. Abdikadirov, Removable Singularities of Separately Harmonic Functions, J. Sib. Fed. Univ. Math. Phys., 2021, 14(3), 369-375. DOI: 10.17516/1997-1397-2021-14-3-369-375.

The theorem on removal of compact singularities (see [1,2]) is one of the most important results in the theory of functions in several complex variables: if a function f is holomorphic everywhere in the domain Q C Cn (n > 1) except a set K C Q, which does not divide the domain (i.e. such that Q\K is connected), then f can be extended holomorphically to whole domain Q. In the work [3], an analogue of this theorem was proved for separately harmonic functions, i.e. for functions which are harmonic in each variable separately: let D be a domain in R"(i) x Rm(y), n,m> 1, K C D a compact set such that D \ K is connected. If the function u(x,y) is separately harmonic in D \ K, then it harmonically continues to D.

1. Separately harmonic functions

Definition 1. If a function u(x,y) is defined in the domain D C Rn(x) x Rm(y) and satisfies the following properties:

1) for any fixed x0 : {x = x0} n D = 0, a function u(x0, y) is harmonic in y on {x = x0} n D;

2) for any fixed y0 : {y = y0} n D = 0, a function u(x, y0) is harmonic in x on {y = y0} n D, then it is called a separately harmonic function in the domain D.

One of the main methods of studying extension of harmonic functions is the transition to holomorphic functions, and then using the principles of holomorphic extensions.

*[email protected] tabdikadirov1983<8 inbox.ru © Siberian Federal University. All rights reserved

Lemma 1 ([5]). For any domain D C R"(i) C Cn there is a domain of holomorphy D C Cn(z) such that D C D and any harmonic function u(x) in D holomorphically extends into the domain D, i.e. there is a holomorphic function fu(z) in D such that fu \D = u.

The existence of the domain D follows easily from the representation of harmonic functions by the Poisson integral. Indeed, let B = B(x0, R) c D be an arbitrary ball in D, and u(x) be a harmonic function in D. Then the following formula holds

( ) 1 r R2 - \x - x0\2 u(x) = -J R\x - y\n u(y)da(y)

dB

where an is the surface area of the unit sphere. It is clear that the Poisson kernel

1 R2 - |x - x0|2

p (x,y) =

an R\x - y\r

for any fixed y G dB holomorphically extends to some domain B G Cn, B D B. Eventually, B is a Lie ball centered at x0 = (x0,x2,... ,x^) with the radius R (see [11])

B =4 z e C

n

\

\z - x0f +

\z - x0f -

E Z - x°°)

j=l

2

< R

Consequently, every harmonic in B function holomorphically extends to B, which implies the existence a domain D, D C D C Cn satisfying the above properties.

It can be seen from the construction that for each fixed z0 £ D there is a constant Mzo such that

\fu (z0)\ < Mz0\\u\\D, (1)

nevertheless, Mzo is bounded on compact subsets of D and

lim Mz = 1.

z^xeD z

2. Separately analytic functions

Let two domains D C Cn, G C Cm and two subsets, E C D, F C G be given. Assume that a function f (z,w), determined firstly on the set E x F, has the following properties:

a) for any fixed w0 G F, a function f (z,w0) holomorphically extends to the domain D;

b) for any fixed z0 £ E, a function f (z0,w) holomorphically extends to the domain G.

In this case f (z,w) defines some function on the set X = (D x F) U (E x G) and it is called a separately-analytic function on X.

We will use the following theorem on analytic continuation of separately-analytic functions (V. Zakharyuta [8], J. Sichak [9], and see also [7]): let two domains D C Cn, G C Cm be strongly pseudoconvex and two subsets E C D, F C G be non-pluripolar Borel sets. If f (z,w) is a separately analytic function on the set X = (D x F) U (E x G), then it extends holomorphically to the domain

X = {(z,w) G D x G : u*(z,E, D) + u*(w,F, D) < 1}.

2

Here w*(z,E, D) is the P-measure of the set E with respect to the domain D (see [7,8,10]). It is defined as an extremal plurisubharmonic function

u*(z,E, D) = Jim E, D),

where

u(z, E, D) = sup{u(z) : u e psh(D),u\0 < 1, u\E < 0}.

3. On Lelong's theorem

P. Lelong [4] proved the following analogue of the fundamental theorem of Hartogs (see [1], Ch. 1): if u(x,y) is separately harmonic in the domain D C Rn(x) x Rm(y), then it is harmonic in D in both variables.

The proof of Lelong's theorem can be obtained easily if we use the above theorem of V. Zakharyuta and J.Sichak: if u(x,y) is separately harmonic in the domain D C Rn x Rm and B1 C Rn, B2 C Rm are arbitrary balls such that B1 x B2 C D, then by Lemma 1 it extends to the set X = (B1 x B2) U (Bi x B2) as a separately analytic function. Therefore, u(x, y) extends holomorphically to the domain

X = {(z,w) e B1 x B2 : u*(z,B1,B1) + u*(w,B2,B2) < 1} .

Since B1 x B2 C X, the function u(x, y) is infinitely differentiable in B1 x B2 and therefore, harmonic in both variables. Since the balls are arbitrary, it follows that u(x, y) is harmonic in both variables in the domain D.

4. The main results

Now we are ready to prove the main results of this paper.

Theorem 1. Let S be a closed subset of the domain D C Rn(x) x Rm(y), n,m> 1, and its orthogonal projections S1 = {x e Rn : (x,y) e S} and S2 = {y e Rm : (x,y) e S} are nowhere dense. Then any function u(x, y) which is separately harmonic in the domain D \ S extends harmonically to the domain D.

Proof. Let u(x, y) be a separately harmonic function in the domain D \ S and the projections of the closed set S:

S1 = {x e Rn : (x, y) e S} ,S2 = {y e Rm : (x, y) e S} ,

are nowhere dense. We denote by S C S the set of non-removable singularities for the function u(x,y). Suppose that S = 0. We take arbitrary balls B1 C Rn(x) and B2 C Rm(y) such that B1 x B2 C D and (B1 x B2) n S = 0. We denote by

S1 = {x e B1 : (x, y) e (B1 x B2) n S}, S2 = {y e B2: (x, y) e (B1 x B2) n S}.

Since (B1 x B2) n S C S1 x S2, we have

(B1 x B2) \ (S1 x S2) = (B1 x (B2 \ S2)) U ({B1 \ S1) x B2) C (B1 x B2) \ S.

Hence, by Lemma 1, the function u(x,y) can be extended analytically to the set X = = (^Bi x (B2 \ S2)) U (^(Bi \ Si) x B2 j as a separately analytic function. Consequently, u(x,y) extends holomorphically to the domain

X ={(z,w) G Bi x B2 : u*(z,Bi \ Si,Bi) + u*(w,B2 \ S2B2) < l} . Since the sets Bi \ Si, B2 \ S2 are locally pluri-regular, we get

X C X, i.e. (Bi x B2) \ (Si x S2) C X.

(About pluri-regular sets and their properties, see [6,12]). Now we take an arbitrary point a G Si and x0 G U(a, e) \ Si, where U(a,e) = {x : \x - a\ < e}, 0 < e < ^ dist(a, dBi). For the point x0 there is a point a0 G Si such that

00

d = \x0 - a0\ = inf {\x0 - x\ : x G ¿^j .

It is clear that the intersection Bi n {x : \x° - x\ < d} C Bi \ S\ contains the interval (x0, a0), which is not pluri-thin at the point a0 G Si (see [6], Proposition 4.1). Hence, it follows that

u*(a°,Bi \ Si,Bi) =0.

On the other hand, there is a point b° G S2 such that (a0, b0) G S and by the definition of P-measure there is also some number 62 : w*(b0, B2 \ S2,B2) < S2 < 1. Now we take some number ^ > 0 so that + S2 < 1. Hence, an open neighborhood of the point

(a0,b0) G S:{ z : w*(z,Bi \ SSi,Bi) < x {w : w*(w,B2 \ ¡S2,B2) < ,

is contained in X, i.e. the point (a0, b0) G S is a removable singularity and this contradicts our assumption concerning S. Thus S = 0. The theorem is proved. □

Using methods of V. Zahariuta on analytic extension of separately analytic functions we get the following result which generalizes Hamano's theorems [3].

Theorem 2. Let two domains D C Rn, G C Rm and two sets E C D, F C G be given. If E c D is compact and F is a closed subset of G with nonempty complement G \ F = then any separately harmonic function u(x, y) in (D x G) \ (E x F) harmonically extends to the domain D x G.

Proof. According to Lemma 1 there is a pseudoconvex domain G C Cm such that G C G and for each fixed x G D \ E a function u(x, •) holomorphically extends to G. Moreover, there is a sequence of strongly pseudoconvex domains Gj,j = 1, 2,... such that Gj c Gj+i c G,

^ 00 ^ ^

G = U (Gj and (G n Gi) \ F = 0. According to (1) for the set

j=i

K£ = {z G D : dist(z, E) < e} c D, where e > 0 is a small enough number, there is a sequence of positive real numbers Mj such that

\u(x,w)\ < Mj y(x,w) G dKe x Gj+i.

Consequently, for any l e N there is a sequence of positive numbers N(l) such that the inequality

E

\a\<l \JGj

dlalu(x, w)

dwa

2

dV) < j Vx e dKe (2)

holds.

Now we take a closed ball B C (G n <G1) \ F and for a fixed j and a sequence of sets B C Gj we consider a Hilbert space H0 C H1. For H0 we take the closure of the space

O(G) n h(G) n W2(Gj), l > m.

(Here O(G) is the space of holomorphic functions on G, h(G) is the space of harmonic functions on G and W2(Gj) is the Sobolev space.) For H1 we take the closure of the space h(G) nL2(B,a), where 1

L2 (B,V)=i^ f \f (W)\2d^ 2 ^

and d,a = ^ddcw*(w, B,Gj)Xl (see [7,8,10]). Let {ek( w)}f=1 be the common orthogonal basis

for spaces H0 C H1 such that ||ek||Ho = , \\ek\\Hl = 1, mkm ^ ln^k ^ Mkm, and M is a constant, k = 1,2,... (see [8,13]).

From the continuous embedding of H0 C C(Gj) n O(Gj) it follows that

\ek(w)\ < C\ek\Ho = C^k, w e Gj, (3)

where C is a constant.

We consider the set Ak = {z e B : \ek(y)\ > k}. By Chebyshev's inequality we have

1 I" 1 1

a(Ak) < J2 B \ek(y)\2d°(y) = k2 Mm = p, k = 1, 2,....

CO / CO \ _ CO CO

Consequently, £ a(Ak) < to and lim ct |J Ak) = 0. We let Us = B \ |J Ak, U = U Us.

k=1 \k=s J k=s s = 1

Then a(B \ U) = 0. Therefore, u*(w,B,Gj) = u*(w,U,Gj), i.e. w*(w,Us,(Gj) | u*(w,B,Gj), w e Gj (see [7,10]). Since \ek(y)\ < k, w e Es, k > s, taking into account (3), by two constants theorem we obtain the following estimation

\ek(w)\ < c(s)k^k'{w'Us'Gjk > s, w e Gj, (4)

where c(s) is a constant independent of k.

Now we compare the formal Fourier-Hartogs series to the function u(x, w), (x, w) C D x Gj,

oo

u(x, w) ak(x)ek (w), (5)

k = 1

where the coefficients are defined by the usual formulas of the space H1:

ak (x) = u(x,w)ek (w)da, k = 1, 2,....

Jb

We show that the series (5) converges locally uniformly in the set Ke x Gj.

2

Since the function u(x,y) is continuous and separately harmonic on the set D x B, it follows that ak(x) is harmonic on D. Moreover, for any fixed x G dKe the function u(x,w) G H0, then

\ak(x)\ = (u(x, •),ek)hi = V-2(u(x, •),ek)Ho. Consequently,

\au(x)\ < -1\\u(x, OH Ho\\ek || Ho < llu(X, ^ , x G dKe. Vk Vk

Hence, by the estimation (2) and the maximum principle we get the following estimation

Nl

\ak (x)\ < ,k =1, 2,..., x G Ke. (6)

Vk

Comparing the estimates (4) and (6), we obtain

\ak(x)ek(w)\ < c(s)Njkvk < c(s)Njke ,

k > s, (x, w) G Ke x Gj, where Us C B, a(Us) > 0. The last estimation shows that the series (5) converges locally uniformly on the set Ke x Gj and its sum u(x, w) coincides with u(x, w) on the set dKe x Gj, i.e. u(x, w) is an analytic continuation of u(x, w). Finally, letting j tend to infinity we obtain an analytic continuation of the function u(x,w) on the set Ke x G which contains the set E x F, that is the function u(x, y) can be separately harmonically extended to D x G. The proof of Theorem 2 is completed. □

Comparing the ideas of proof of theorems above, one can easily prove the following theorem:

Theorem 3. Let two domains D C Rn, G C Mm and two sets E C D, F C G be given. If E is a nowhere dense closed subset of the domain D and F is a closed subset of the domain G with a non-empty complement G \ F = 0, then any separately harmonic function u(x, y) on the domain (D x G) \ (E x F) can be extended harmonically to the domain D x G.

References

[1] B.V.Shabat, Introduction to Complex Analysis, Part II, Moscow, Nauka, Fiz. Mat. Lit., 1985 (in Russian).

[2] R.Gunning, H.Rossi, Analytic functions of several complex variables, Chelsea Publishing, American Mathematical Society, Providence, Rhode Island, 1965.

[3] S. Hamano, Hartogs-Osgud theorem for separately harmonic functions, Proc. Japan Acad. A, 83(2007), 16-18. DOI: 10.3792/pjaa.83.16

[4] P.Lelong, Fonctions plurisousharmoniques et fonctions analytiques de variables reelles, Ann. Inst. Fourier. Paris., 11(1961), 515-562.

[5] A.Sadullaev, S.A.Imomkulov, Extension of holomorphic and pluriharmonic functions with subtle singularities on parallel sections, Proc. Steklov Inst. Math., 253(2006), 144-159. DOI: 10.1134/S0081543806020131

[6] A.Sadullaev, Plurisubharmonic measures and capacities on complex manifolds, UMN, 36(4)(1981), 53-105.

[7] A.Sadullaev, Plurisubharmonic functions, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr., 8(1985), 65-111 (in Russian).

[8] V.Zakharyuta, Separate-analytic functions, generalized Hartogs theorems and hulls of holo-morphy, Math. Coll., 101(1976), no. 1, 57-76.

[9] J.SiCiak, Separately analytic functions and envelopes of holomorphy of some lowerdimen-sional subsets of Cn, Ann. Pol. Math., 22(1969), no, 1, 145-171.

[10] E.Bedford, B.A.Taylor, A new capacity for plurisubharmonic functions, Acta. Math., 149(1982), 1-40.

[11] V.Avanissian, Cellule d'Harmonicite et Prolongement Analitique Complexe, Travaux en cours, Hermann, Paris, 1985.

[12] W.Plesniak, A criterion of the L-regulerity of compact sets in Cn, Zeszyty Neuk. Uniw. Jagiello, 21(1979), 97-103.

[13] B.Mityagin, Approximate dimension and bases in nuclear spaces, Russian Math. Surveys, 16(1961), no, 4, 59-127.

Стираемые особенности сепаратно-гармонических функций

Севдиёр А. Имомкулов

Хорезмский областной филиал Математического института им. В. И. Романовского

Академия наук Республики Узбекистан Ургенч, Узбекистан

Султанбай М. Абдикадиров

Каракалпакский государственный университет

Нукус, Узбекистан

Аннотация. В работе рассматриваются устранимые особенности сепаратно-гармонических функций. Точнее, доказана теорема о гармоническом продолжении сеператно-гармонической в Б \ Я функции и(х,у) в область Б, где Б С К"(х) х Кт(у), п,т > 1 и Я — замкнутое подмножество области Б, а ее проекции Яг = {х £ К" : (х,у) £ Я} и Я2 = {у £ Кт : (х,у) £ Я} нигде не плотны.

Ключевые слова: сепаратно-гармоническая функция, псевдовыпуклая область, интеграл Пуассона, Р-мера.