Stationary harmonic functions on homogeneous spaces Текст научной статьи по специальности «Математика»

ISSN 2074-1863 Уфимский математический журнал. Том 7. № 4 (2015). С. 155-159.

УДК 517.572

STATIONARY HARMONIC FUNCTIONS ON HOMOGENEOUS SPACES

V.S. KHOROSHCHAK, A.A. KONDRATYUK

Abstract. Stationary harmonic functions on homogeneous spaces are considered. A relation to double periodic harmonic functions of three variables is showed.

Keywords: harmonic function, multiplicatively periodic function, double periodic function, homogeneous space, Klein space, invariant family, stationary element with respect to a subgroup, punctured Euclidean space.

Mathematics Subject Classification: 31B05

1. Introduction

A function f on C* = C\{0} is said to be multiplicatively periodic if there exists q, 0 < |q| = 1 such that

Vz e C* f (qz) = f (z). (1)

Such a q is called multiplicator of f.

The theory of meromorphic functions satisfying (1) is dual to the theory of elliptic functions, which are double periodic meromorphic functions on C ([1]-[3]).

Holomorphic functions, harmonic functions, subharmonic functions satisfying (1) are constant.

In this connection we try to answer the questions:

(i) do multiplicatively periodic non-constant harmonic functions of several variables exist?

(ii) do double periodic non-constant harmonic functions of three variables exist?

(iii) if yes, what are their representations?

Note that C* is a nonlinear homogeneous space on which multiplicative group C* acts and that (1) implies

Vn e Z Vz e C* f (qnz) = f (z).

We will say that f is stationary with respect to the cyclic group {qn}, n e Z, generated by q. Note also that each multiplicatively periodic harmonic functions of multiplicator q, 0 < q = 1,

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in the punctured Euclidean space Rm = Rm\{0}, m > 3, is constant due to the extremum principle and the counterpart of the Liouville theorem. Hence, in order to solve problems (i)-(iii) we should consider more general homogeneous spaces.

2. Homogeneous spaces

Definition 1. Let X be a topological space, G be a group of homeomorphic mappings of X onto X. A couple (X,G) is called homogeneous space.

V.S. Khoroshchak, A. A. Kondratyuk, Stationary harmonic functions on homogeneous spaces.

© Khoroshchak V.S., Kondratyuk A.A. 2015.

If G is transitive, that is

Vxi,x2 E X 3t E G (x2 = txi), then (X, G) is said to be the Klein space (see [4]).

Example 1. Let G be a group of the linear transformations of Euclidean space Rn. It is transitive. Then (Rn,G) is the linear homogeneous space. It is the Klein space.

Example 2. Let X = Rn\{0}, G = SO(n). X is invariant with respect to homothetic transformations, G is intransitive. Thus, instead of G the compositions of rotations and homothetic transformations G1 can be taken.Then G1 is transitive and (Rn\{0},G1) is the nonlinear homogeneous space. It is the Klein space, too.

3. Invariant functional spaces on (X, G). Stationary elements with respect

to subgroups

Definition 2. Let (X, G) be a homogeneous space. A set (family) of functions F is said to be invariant if it satisfies the following condition

Vf EF Vt E G (f o t eF).

Definition 3. Let (X, G) be a homogeneous space, F be an invariant family, H be a subgroup of group G. An element f E F is called stationary with respect to H if

Vt E H (f o t = f). The set of such elements is denoted by FH.

4. Stationary harmonic functions on homogeneous spaces

The space R3 = {(x1,x2,x3) = x : x2 + x2 > 0} is called pierced Euclidean space. It is nonlinear, invariant with respect to the rotations around axis x3 and homothetic transformations. The composition of the rotations around axis x3 and homothetic transformations forms a group, which we denote by G. Hence, we obtain nonlinear homogeneous

space (R3, G). It is the Klein space.

One of the functional spaces invariant with respect to group G is the linear space of harmonic

in R3 functions (see, for example, [5]). We denote it by H.

The rotation by an angle a around axis x3 is given by the following matrix

(cos a — sin a 0 sin a cos a 0 0 0 1

Fix q, 0 < q < 1. Let H be a composition of some A and the cyclic group {qn}, n E Z.

It is an open problem to describe stationary elements from HH which are harmonic in R3 functions satisfying the condition

Vx E R 3 h(qnAx) = h(x).

However, we can show that class HH is non-trivial, i.e., it contains non-constant harmonic functions.

We consider the series

i i i

— - i —

-^vlal Iqnx — a\) ^ \q-nx — a\

n=0 1 1 1 n=1 1

introduced in ([6]). It was proved there that for any fixed a e R3, q< |a | ^ 1, the remainder

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of series (2) converges uniformly on the compact subsets from R 3 and that the sum K(x, a) of

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(2) is subharmonic in R 3.

Denote X3 = {(0, 0,x3) : x3 = 0}. If a e X3, then each term of (2) is a harmonic function

in R3. Indeed, the fundamental solution of the Laplace equation in R3 is щ. Since qkx = a,

k e Z, the function |qfeX_a| is harmonic in R3. Therefore, the function K(x,a) is harmonic in

R3 if a = (0, 0,a3) e X3, q < |a| ^ 1.

Note that the function K(x, a) is independent of a, namely,

VA K (Ax, a) = K (x, a). (3)

Let y = Ax, that is yi = xi cos a — x2 sin a, y2 = xi sin a + x2 cos a, y3 = x3. Consider the absolute value |qky — a|, k e Z. We have

|qky — a| = | (qkxi cos a — qkx2 sin a, qkxi sin a + qkx2 cos a, qkx3 — a3) | =

= \J q2k xf + q2k x2 + (qk x3 — a3)2 = |qkx — a|.

Thus, identity (3) is valid. It is easy to check that

K(qx,a) = K(x,a) — —. (4)

|a|

Let a = (0, 0,1) and b = (0, 0, —1). The function

h(x) = K(x, a) — K(x, b)

is harmonic in R 3.

Using identities (3) and (4), we obtain

h(qAx) = K(x, a) — — — K(x, b) + —.

|a| |b|

Since |a| = |b|, we get

Vx e R3 h(qAx) = h(x).

Thus, h e .

5. A class of functions in

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We denote by B the class of bounded Borel sets in R3 whose closures belong to R3. For B e B we let

qB = {qx : x e B}, 0 < q < 1.

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Theorem A ([7]). A measure ^ in R 3 is the Riesz measure of a multiplicatively periodic i-subharmonic functions of multiplicator q if and only if

(i) ^(qB) = q^(B) for each B e B;

(ii) f g = 0 for all r > 0.

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Theorem B ([7]). Each multiplicatively periodic i-subharmonic in R3 function u of multiplicator q satisfies the representation

u(x) = C + J K (x,a)duu(a),

q<|a|<i

where C is a constant.

The following theorem describes a class of harmonic functions from Hh.

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Theorem 1. If a Borel measure » on R3 satisfies the conditions

1) »(qB) = q»(B) for each B E B;

2) I ft = 0;

q<M<1

3) »(B) = »(BC\X3), ^(0) = 0;

then the function

h(x) = J K (x,a)d»(a) (5)

q<M<1

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belongs to HH and vice versa each h E HH which admits a 8-subharmonic continuation on R 3 satisfies the representation

h(x) = C + J K (x,a)d»(a), (6)

q<M<1

where C is a constant and » satisfies 1)-3).

Proof. Let » satisfy Conditions 1), 2). According to Theorem B, function h defined by (5) is

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multiplicatively periodic i-subharmonic in R 3 of multiplicator q. In virtue of condition 3) h is harmonic in R3. Taking into account that K(x,a) is independent of A we have h E HH.

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Now let h be a function from Hh admitting a i-subharmonic continuation on R 3. According to Theorems A and B it has representation (6), where » satisfies Conditions 1), 2). Since Ah = 0

in R 3, » satisfies also Condition 3). This completes the proof. □

6. Double periodic harmonic functions in a layer

Let h(x) be a multiplicatively periodic harmonic in R 3 function of multiplicator q, 0 < q = 1. Consider the mapping

x1 = e? cos rq, x2 = e? sin rq, x3 = e? cot Z, where E R, 0 < ( < n. We have x1 + x2 = e2? > 0. Hence, it maps the layer {(Z,V,Z) : E R, 0 < ( < n} onto R 3 with the Jacobians J = . Laplacian A becomes

( d2 d2 d2 (1 d d2 W

A = m? + w+sin2 zd(i+s'n2H 1 * + dddc))-

Denote

g(Z, V, Z) = h(e^ cos rq, e? sin rq, e? cot Z). The function g is defined in the layer {(Z,V,Z) : Z,V E R, 0 < Z < n}. Since h(qx) = h(x), we have

g(Z + log q,v + 2n,Z) = g(Z,v,Z).

Indeed,

g(C + logq, n + 2n, Z) = h(e?+logq cos(n + 2n), e?+logq sin(n + 2n), e?+logq cot Z) =

= hiqe^ cos n, qe? sin n, qe? cot Z) = hie^ cos n, e? sin n, e? cot Z) = g(Z, V, Z). Denoting u1 = log q, u2 = 2n, we obtain double periodic harmonic function g of period A = (Zu1, Zu2,Z). That is, such a function is stationary with respect to a group of the translations indicated above.

Remark. The connection between the local spherical coordinates and the new substitution is as follows

e^ = r sin 0, n = 0, Z = REFERENCES

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4. B.A. Rozenfeld Noneuclidean spaces. Nauka, Moscow, 1969, 548p.(in Russian)

5. E.M. Stein, G. Weiss Introduction to Fourier analysis on Euclidean spaces. Princeton, New Jersey, Princeton University Press, 1974, 332p. (in Russian)

6. A.A. Kondratyuk, V.S. Zaborovska Multiplicatively periodic subharmonic functions in the punctured Euclidean space // Mat. Stud. 40 (2013). P. 159-164.

7. V.S. Khoroshchak, A.A. Kondratyuk The Riesz measures and a representation of multiplicatively periodic ö-subharmonic functions in a punctured Euclidean space // Mat. Stud., 43 (2015). P. 61-65.

Vasylyna Stepanivna Khoroshchak, Ivan Franko National University of Lviv, 1, Universytetska St., 79000, Lviv, Ukraine E-mail: vasylyna1992@rambler.ru

Andriy Andriyovych Kondratyuk,

Ivan Franko National University of Lviv,

1, Universytetska St.,

79000, Lviv, Ukraine

E-mail: kond@franko.lviv.ua