Математичнi Студiї. Т.35, Matematychni Studii. V.35, No. УДК 57.574 O. P. Gnatiuk, A. A. Kondatyuk SUBHARMONIC FUNCTIONS AND ELECTRIC FIELDS IN BALL LAYERS. II O. P. Gnatiuk, A. A. Kondatyuk. Subhamonic function and electic field in ball laye. II, Mat. Stud. 35 20, 50 59. In thi equel to [] we tudy a pecial cae BL,, >. Alo the explicit epeentation of a ubhamonic extenion fo a ubhamonic function ux nea a emovable point i obtained. Moeove, the divee Nevanlinna chaacteitic ae compaed. О. П. Гнатюк, А. А. Кондратюк. Субгармоничиские функции и электрические поля в шаровых прослойках. II // Мат. Студiї. 20. Т.35,. C.50 59. В этом продолжении к [] мы изучаем особый случай BL,, >. Также получено представление в явной форме субгармонического продолжения субгармонической функции ux в окрестности устранимой точки. Кроме того сравниваются различные Неванлинновские характеристики. Intoduction. Let D be an open et in the Euclidian pace R m m 3 and F a compact ubet of D. It i a claical eult ee [2], Theoem 5.8, p. 255 that if u i ubhamonic in D \ F and bounded fom above and moeove F i pola, then u ha a ubhamonic extenion to the whole of D. Gadine [3] ha hown that, in the cae of a compact exceptional et the above boundedne condition can be elaxed by impoing cetain moothne and Haudoff meaue condition on the et. Riihentau [4] ha eplaced the moothne and Haudoff meaue condition with one ole condition on Minkowki uppe content. It ha been etablihed in [5] that if u i a ubhamonic function in D\F and bounded fom above, whee F i a cloed pola et, then the function ux, x D\F, ũx = uy, x F, y x, y / F i ubhamonic in D. In thi equel to [] we give an explicit epeentation of a ubhamonic extenion of u nea a emovable point. The appoach peented hee appea to be new. It allow u to give a compaion of T +0, ; u and T, u fo ubhamonic function in a ball. In thi pape we alo invetigate ubhamonic function on ymmetic ball laye. We ue eult of the fit pat []. 4. Subhamonic function on ymmetic ball laye. Conide a ubhamonic function u on BL,, non identical, and define N 0 ; u:=m 2 200 Mathematic Subject Claification: 3B05. nt m 2 dt tm m 2 nt dt, tm c O. P. Gnatiuk, A. A. Kondatyuk, 20
SUBHARMONIC FUNCTIONS IN BALL LAYERS. II 5 whee nt i the ditibution function of the Riez meaue µ of the function u ee Definition fom []. Coollay. Let u be a ubhamonic function, non identical, in BL, and µ be it Riez meaue. Then N 0, u = uxdσx + 2 m m c m uxdσx m S0, S0, + 2 m uxdσx, > S0, whee i the aea of the unit phee and dσx i an element of the uface aea. Definition. Let u be a ubhamonic function in BL 0, 0, non identical. The function T 0, u:=t, ; u = u + xdσx + 2 m m c + m xdσx m + 2 m S0, S0, S0, u u + xdσx, < < 0 2 i called the Nevanlinna chaacteitic of u, whee u + = max{u; 0}. Like to it countepat fo ubhamonic function in a ball ee [2], p. 46, the Nevanlinna chaacteitic T 0, u ha elementay popetie that have been collected in the following theoem ee [6]: Theoem. Let u, u, u 2 be ubhamonic function in BL 0, 0, non identical. Then a T 0, u i nonnegative, nondeeaing and convex with epect to 2 m, < < 0 and T 0, u = 0. b if u i contant then T 0, u, < < 0 i identical zeo; c T 0, u + u 2 T 0, u + T 0, u 2 + O, 0. T 0, λu = λt 0, u fo λ > 0, < < 0. Poof. Since u + i ubhamonic and accoding to, we can ewite 2 a follow T 0 ; u = N 0 ; u +. The function N 0 ; u + i nonnegative, nondeceaing and convex with epect to 2 m [2]. Hence, T 0, u atifie popety a. The tatement b immediately follow fom Definition. Popety c follow fom the inequality u + u 2 + u + + u + 2. Define m 0, u = m S0, u xdσx + 2 m c m m S0, u xdσx, < < 0, whee u = min{u; 0}. Now we can ewite 2 a follow
52 O. P. GNATIUK, A. A. KONDRATYUK Theoem 2. If u i a ubhamonic function in BL 0, 0 then T 0, u = m 0, u + N 0, u + 2 m u xdσx, < < 0. S0, Thi i a countepat of the fit fundamental theoem fo ubhamonic function on ymmetic ball laye. Moeove, we give a compaion of T 0, u with T, u fo function ubhamonic in a ball. If a ubhamonic function u in BL 0, 0 ha a ubhamonic continuation into B0, 0 = {x: x 0 }, 0 >, then it claical Nevanlinna chaacteitic T, u i alo detemined. By a ubhamonic continuation we mean that thee exit a ubhamonic function u in B0, 0 with Riez meaue µ uch that u = u on BL 0, 0 and the etiction of µ on BL 0, 0 i equal to µ, i.e. µ {x: t < x t} = µ {x: t < x t}, < t < 0, whee µ i the Riez meaue of u. Then, uing, we have N 0, u = N, u + N 2 m, u + 2 m N, u, < < 0, 3 whee N, u i defined in [2]. Moeove, T 0, u = T, u + 2 m T, u + 2 m T, u, < < 0. 4 Equality 4 follow fom the definition of T 0, u and T, u ee [2] immediately. Coollay 2. Let ux be a ubhamonic function, non identical, in BL, and let µ be the Riez meaue of ux. Then fo ξ BL, uξ = uxp m ξ, xdσx+ S0, + c m uxp ξ, xdσx Gξ, xdµx, m S0, BL, whee and P ξ, x = n=0 P ξ, x = ξ 2 m Gξ, x = G ξ, x n=0 2n + m 2 n+m 2 ξ 2n+m 2 m 2 ξ 22n+m 2 pν nco φ, n=0 2n + m 2 n 2n+m 2 ξ 2n+m 2 p ν m 2 ξ 22n+m 2 nco φ, 22n+m 2 x n+m 2 x n ξ n+m 2 ξ n whee G x, ξ i the Geen function in the ball of adiu centeed at the oigin. The convegence i unifom on compact ubet of BL,. Set B 0, u = B, ; u, i.e, B 0, u = max{m ; u; M; u}, > whee Mt; u = up{ux: x = t}, t.
SUBHARMONIC FUNCTIONS IN BALL LAYERS. II 53 Coollay 3. If ux i a ubhamonic function in BL,, then fo < ρ < we have + ρ T 0ρ; u B 2 m 0 ρ; u + m 2 + ρ T ρ m 0 ; u + + 2 m S0, u + xdσx. 5 Let S be a eal and nonnegative function inceaing fo 0 < <, whee 0 > 0. The the ode λ and the lowe ode µ of the function S ae defined a λ = up log S log, µ = log S inf log, Obviouly, the ode and the lowe ode of the function atify the elation 0 µ λ. Definition 2. If ux i a ubhamonic function in R m \ {0}, then the ode λ and the lowe ode µ of u ae called the ode and the lowe ode of T 0, u. Theoem 3. If ux i a ubhamonic function in R m \ {0}, the ode λ and the lowe ode µ of the function T 0 ; u and B 0 ; u ae the ame, i.e., λ[u] = λ 0 [u], µ[u] = µ 0 [u], whee λ[u] = up log T 0, u, λ 0 [u] = up log B 0, u, log log µ[u] = inf log T 0, µ 0 [u] = inf log B 0. log log Poof. The concluion follow fom Coollay 3. Indeed, etting in 5 = γρ, γ >, povided u + i poitive in x = ρ fo the cetain ρ, we get T 0 ; u + ρ B 0ρ; u γm 2 + γ T 2 m γ m 0 γρ; u + + γρ2 m S0, u + xdσx. 6 Fom 6 we deduce at once that λ[u] = λ 0 [u] and µ[u] = µ 0 [u], ince the ode and the lowe ode of T 0 ; u i not geate than that of B 0 ; u and alo that the ode and the lowe ode of B 0 ; u i not geate than that of T 0 ; u, completing the poof. Coollay 4. If ux i a ubhamonic function of finite lowe ode µ in R m \ {0}, then inf B 0; u T 0 ; u Kµ, m, whee m µ 2 expµ 2m exp Kµ, m 3, µ m, Kµ, m, 0 < µ < m m µ and K0, m =. The poof i imila to that of Theoem 4.3 in [2], p.66. 5. Repeentation of a ubhamonic extenion of a ubhamonic function nea a emovable point. Let BL, be the annula egion {x R m : < x < }, 0 < <.
54 O. P. GNATIUK, A. A. KONDRATYUK Definition 3. Let D be an open ubet of R m. If x 0 D and u i a ubhamonic function in D\{x 0 } then u i aid to have an iolated ingulaity at the point x 0. Definition 4. The ingula point x 0 of a ubhamonic function u i emovable if thee exit a ubhamonic function v in D that coincide with u fo all x D\{x 0 }. We ay that v i a ubhamonic extenion of u. Theoem 4. Subhamonic function u in D ha a emovable ingulaity at x 0 D if and only if thee exit a ubet {x: 0 < x x 0 < } D on which u i bounded fom above. Futhemoe, the extenion v of u can be epeented nea the emovable point in the following way vx = uξp x, ξdσξ Gx, ξdµξ γ x 2 m 7 Sx 0, 0< ξ < whee P x, ξ and Gx, ξ ae Poion kenel and Geen function fo the unit ball epectively, γ = m 2 I; u, 0 γ <, I; u = uξdσξ. 8 x0 m Recall that Bx 0, = {x: x x 0 < }. Sx 0, Poof. By Definition 4 and the popety of a ubhamonic function, the extenion v of u i bounded fom above on ome compact ubet of Bx 0,. Since u coincide with v on {x: 0 < x x 0 < }, u i bounded fom above thee. Uing linea tanfomation that peeve ubhamonicity, x x 0 + 2 x one can get a ubhamonic function u in BL0, 2. Since u i bounded fom above, thee exit a contant C uch that u C. Without lo of geneality, we can conide the function u C intead of u. Let µ be the Riez meaue of u on BL0, 2. We extend µ to B0, in uch a way { µe 0 / E νe = 9 µ + µ 2 0 E, whee µ = µe\{0} and µ 2 = γδ0, δ0 i the Diac delta-function, and E i a Boel et. In the cae 0 / E the meaue µe i finite a the Riez meaue. In the cae 0 E we have the um of two meaue. Since γ <, the econd meaue i finite. Indeed, if we have a convex function ft on uch an inteval a, +, then thee exit uch a it ee [2], p.3 ft < +. t t Uing the ubtitution t = 2 m, we obtain 8. Note that I; u i a convex function with epect to 2 m [2], Theoem 2.2, p.8. Now we pove that µe\{0} <. It will be enough to pove that µbl0, <, becaue E\{0} BL0,.
SUBHARMONIC FUNCTIONS IN BALL LAYERS. II 55 Uing Definition fom [], we can et n = 0. Hence, we have µbl0, = n n0 = n0. Fixing = 0 in Theoem 3 fom [] and conideing the it a tend to 0, we get 0 m 2 nt t dt C m m 2 fo ome contant C that doe not depend on. Suppoe that n0 ; then thee exit a equence { k } of k N. In view of thi and integating by pat, we obtain m 2 A 0, we get nt k nt dt m 2 tm t dt m m 2 k 0 m 2 nt t k m m 2. That contadict 0. Hence, n0 >, i.e., µbl0, <. Next we conide the function v = u P Gdν, B dt t = k m m 2 + k m 2 m 2 m 2 k. 0 that i ubhamonic in B0, and u P i the convolution of u with the Poion kenel on B0,. Uing 9 fo all x B0,, we have vx = uξp x, ξdσξ Gx, ξdµξ γ x 2 m. Sx 0, 0< ξ < It emain to how that u = v fo all x BL0,. Let u chooe uch that 0 < < and fix x. Applying Poion-Jenen Theoem 2 fom [] to a ubhamonic function u in BL,, we obtain ux = S0, S0, + m 2 S0, uξ uξp x, ξdσξ n= m 2 x 2 m m 2 S0, uξdσξ 2n + m 2 2n+m 2 x 2 m n x n p ν m 2 2n+m 2 nco φdσξ+ [ x co φ uξ 2 + x 2 2 x co φ m 2 x 2 m m 2 co φ x x 2 x 2 + 2 x co φ m 2 uξdσξ+ S0, ] dσ
56 O. P. GNATIUK, A. A. KONDRATYUK + + S0, < ξ < < ξ < n= uξ n= Gx, ξdµξ + n + n 3n+m 3 x 2 m n x n p ν m 2 2n+m 2 nco φdσξ m 2 x 2 m m 2 < ξ < ξ 2 m dµξ+ 2n+m 2 x 2 m n x n ξ 2 m n ξ n p ν 2n+m 2 nco φdµξ. Let u calculate the it of the ight-hand ide of a tend to 0. The fit ummand doe not depend on. The next one x 2 m m 2 uξdσξ = 0, 0 m 2 S0, becaue u i bounded fom above. Let u etimate the thid tem. Since u 0 and we get S0, max{p ν nx: x } = C m n+m 3 = uξ n= m 2 x n= chooing < x, we obtain 2 m 2 x n= 2n + m 2 2n+m 2 m 2 2n+m 2 2n + m 2 n nc n m 2 2n+m 2 x 2n + m 2! 2n n!m 2! 2 n + m 3!, n!m 3! x 2 m n x n p ν nco φdσξ n+m 3 S0, S0, uξdσξ. uξdσξ The lat eie convege unifomly. Thu, the thid tem i vanihing a 0. Let u etimate the fit ummand of the foth tem [ ] x co φ uξ dσ m 2 2 + x 2 2 x co φ m 2 S0, x + m 2 m 2 uξ m x dσξ. m S0, Since γ i finite, the it of the ummand i 0. The analogou eaoning deal with the othe ummand of the foth tem. Conide the it of the next addend x 2 m uξdσξ = x 2 m m 2 I; u = 0 m 2 0 m 2 S0, = γ x 2 m.
SUBHARMONIC FUNCTIONS IN BALL LAYERS. II 57 It can be poved in the ame manne a it wa done fo the thid tem that the it of the fifth one i 0. Now we pove that Gx, ξdµξ = Gx, ξdµξ. 2 0 < ξ < 0< ξ < A we know, Geen function ha a ingulaity at the point x = ξ, that i why we et < 0 < x <, and by the additivity popety of Lebegue integal, we have Gx, ξdµξ = Gx, ξdµξ + Gx, ξdµξ. < ξ < < ξ < 0 0 < ξ < By Popoition fom [7], p. 20, < ξ < 0 Gx, ξdµξ = µ{ξ : < ξ < 0 } and by the popety of meaue continuity ee [7], p. 6, thee exit a equence { n } with n N uch that Gx, ξdµξ = Gx, ξdµξ. n n< ξ < 0 0< ξ < 0 Hence, we have poved 2. Now we conide the it of the eighth addend x 2 m m 2 ξ 2 m dµξ = 0 m 2 = x 2 m [ m 2 0 m 2 < ξ < < ξ < < ξ < dµξ m 2 ξ m 2 0 m 2 < ξ < dµξ Since the meaue µbl0, i finite, the lat ummand i vanihing. Conide m 2 ξ 2 m dµξ = m 2 dnt. 3 tm 2 Integating the ight-hand ide of 3 by pat, we obtain The it of 4 a 0 i m 2 n n + m 2 m 2 n0 + m 2 m 2 0 nt t dt. 4 m nt dt. 5 tm ].
58 O. P. GNATIUK, A. A. KONDRATYUK Since m 2 m 2 nt dt = m 2m 2 tm m 2 m 2 n t2 m 2 m nt t m dt + m 2m 2 nt dt tm + m 2 m 2 n t2 m 2 m = n0. We conclude that 5 i 0. Now we etimate the lat tem of 2n+m 2 x 2 m n x n ξ 2 m n ξ n p ν 2n+m 2 nco φdµξ < ξ < n= x m 2 x m 2 0< ξ < n= n= C n n+m 3 n Cn+m 3 n n n dµξ 2n+m 2 x ξ ξ m 2 2 2n 0< ξ < dµξ ξ m 2, <. The lat eie convege unifomly and the lat tem i vanihing. The poof i completed. 6. Compaion of T +0, ; u and T, u fo ubhamonic function extended into a ball. We conide bounded fom above ubhamonic function in BL0,. Accoding to Theoem 3, uch function ae extended to B0,. Uing Theoem 3 fom [] and 8, we get Hence, N, ; u = 0 m 2 0 m 2 = 2 m m S0, 2 m m S0, uxdσx S0, m 2 uxdσx + 0 m 2 m S0, 2 m N+0, ; u + γ = uxdσx m Suppoe u0. Accoding to [2], we have Moeove ee, definition 2 fom [], T, ; u = 0 2 m m S0, S0, uxdσx S0, uxdσx uxdσx = uxdσx γ. S0, 2 m N+0, ; u + γ = N, u N, u. S0, u + xdσx + 0 m 2 m 2 m uxdσx. S0, u + xdσx
SUBHARMONIC FUNCTIONS IN BALL LAYERS. II 59 2 m S0, u + m 2 xdσx 0 m 2 S0, u + xdσx. 6 Since u + i a ubhamonic and bounded fom above function, we apply Theoem 3 to u +. Denote 0 m 2 u + xdσx = γ m. S0, A fa a 0 γ < and u + = max{u; 0}, we have that γ = 0. Thu, 2 m T +0, ; u = T, u T, u. 7 Equality 7 follow fom 6 and fom the definition of T, u ee [2] immediately. Note that 7 i tue in both cae u0 = and u0. REFERENCES. Gnatiuk O.P., Kondatyuk A.A. Subhamonic function in ball laye. I// Mat. Stud. 200. V.34, 2. P. 80 92. 2. Hayman W.K., Kennedy P.B. Subhamonic function Mi, Mocow, 980. in Ruian 3. Gadine S.J. Removable ingulaitie fo ubhamonic function// Pac. J. Math. 99. V.47. P. 7 80. 4. Riihentau J. Removable et fo ubhamonic function// Pac. J. Math. 2000. V.94. P. 99 208. 5. Kek M. Pluipotential theoy. Claendon Pe, 99, 59 p. 6. Kondatyuk A., Laine I. Meomophic function in multiply connected domain, Univ. Joenuu Dept. Math. Rep. Se. 2006. V.0. P. 9. 7. Rudin W. Real and complex analyi, McGaw-Hill, New Yok, 970. Faculty of Mechanic and Mathematic Lviv National Univeity Univeytet ka, 79000, Lviv kond@fanko.lviv.ua okanka.gnatyuk@gmail.com Received 6.09.0