Vladimir Azarin About connection between D -topology and the topology of exceptional sets for subharmonic functions see [9], [1, Ch. 3]. We also use C

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1 Journal of Mathematical Physics, Analysis, Geometry 27, vol. 3, No. 1, pp On the Polynomial Asymptotics of Subharmonic Functions of Finite Order and their Mass Distributions Vladimir Azarin Department of Mathematics and Statistics, Bar-Ilan University Ramat-Gan, 529, Israel azarin@macs.biu.ac.il Received September 1, 26 We obtain the results analogous of those of [5] on the polynomial asymptotics with arbitrary < ρn < ::: < ρ 1 < ρ, defining multipolynomial terms. Key words: subharmonic function, asymptotic representation, limit set for entire and subharmonic functions, topology of distribution. Mathematics Subject Classification 2: 3D2, 3D35, 31A5, 31A1. 1. Introduction and the Main Results 1.1. In papers [1 8], it is considered, in particular, the polynomial asymptotics of subharmonic functions of finite order ρ and their mass distributions in terms of the growth of reminder terms and topology of exceptional sets. Besides, the exponents ρ 1 ;:::;ρ n of terms had to satisfy the conditions [ρ] <ρ n <:::<ρ for a noninteger ρ. We are going to represent another point of view by studying the polynomial asymptotics in D -topology and a little bit stronger topology and relax restriction on the exponent to the natural ρ>ρ 1 > ::: > ρ n >. It occurs that this change of topology together with the consideration of more narrow class than in [5] allows to obtain a multiterm asymptotic analog of Levin Pfluger's theory of completely regular growth and make simpler (in our opinion) formulating of the results and proofs. By D -topology we call the topology of the space D (C n) of distributions (i.e., linear bounded functionals) over the basic space D(C n) of finite infinitely differentiable functions. Recall that a sequence u j!, j!1in this space if the linear functionals <u j ;g >! (1:1:1) for all g 2D: cfl Vladimir Azarin, 27

2 Vladimir Azarin About connection between D -topology and the topology of exceptional sets for subharmonic functions see [9], [1, Ch. 3]. We also use Cq;p*-topology, 1 i.e., the topology of linear functionals over the basic space C 1 q;p with the convergence defined like in (1.1.1). The space C 1 q;p is one of the infinitely differentiable functions in C n that tends to 1 not faster then O(jzj q ) as z! and tends to zero not slower then O(jzj p ) as z!1. Let us note that this topology is stronger that D -topology because C 1 q;p ffd(c n): Let u(z) be a subharmonic function in C of normal type with respect to a finite order ρ, i.e., <ff[u] := lim sup M (r;u)r ρ < 1; r!1 where M (r;u) := max jzj=r u(z).we write u 2 SH(ρ): Let μ be a mass distribution in C with no mass in zero. It has normal type with respect to the exponent ρ if < [μ] := lim sup μ(k r )r ρ < 1; r!1 where K r := fz : jzj <rg: We write μ 2M(ρ): Define by μ u the mass distribution associated with u: Recall Borel's Theorem. Let [ρ] <ρ:if u 2 SH(ρ), then μ 2M(ρ) and vice versa. Let ρ =[ρ] :=p. Set ffi R (z; μ; p) := 1 p j j<r < ρ z μ(dοd ): This is a family of the homogeneous harmonic polynomial of degree p: Recall in an equivalent formulation LindelΞof's Theorem. If u 2 SH(ρ) then μ u 2M(ρ) and the family fffi R g is precompact as R!1; and vice versa. Denote u t (z) :=u(tz)t ρ : The function u(z) 2 SH(ρ) is called a function of the completely regular growth (CRG-function) if u t! h ρ (z) in D -topology, as t!1: Here h ρ (z) :=r ρ h(e iffi ) (1:1:2) and the function h(e iffi ) is a ρ-trigonometrically convex function (ρ-t.c. function) (see, e.g., [1, Ch. 1, 15, 16] ), i.e., it is a 2ß-periodic generalized solution of the equation h + ρ 2 h = (dffi); (1:1:3) where is a 2ß-periodic positive measure. 6 Journal of Mathematical Physics, Analysis, Geometry, 27, vol. 3, No. 1

3 On the Polynomial Asymptotics of Subharmonic Functions of Finite Order Recall also that ρ-t.c. function as a distribution is equivalent to a continuous function and can be represented for noninteger ρ in the form h(ffi) = 1 2ρ sin ßρ 2ß Λ cos ρ(ffi ψ ß) (dψ); (1:1:4) where the function Λ cos ρ(ffi) is a 2ß-periodic extension of the function cos ρffi from the interval ( ß; ß) on ( 1; 1). If ρ(> ) is integer, then must satisfy the condition 2ß and the representation has the form h(ffi) =<fce iffi) g + 1 2ρ e iρffi (dffi) =; (1:1:5) 2ß Λ(ffi ψ) sin ρ(ffi ψ) (dψ); (1:1:6) where C is a complex constant, the function Λψ means the 2ß-periodic continuation of the function f (ψ) :=ψ from the interval [; 2ß) on ( 1; 1): Recall (see [9], [1, Ch.3, 1]) that μ t (do not confuse with μ u ) is the mass distribution defined by the equality <μ t ;g >:= t ρ g(z=t)μ(dxdy) for all g 2D: It can also be defined by the equality μ t (E) :=μ(te)t ρ ; where E is every Borel set and te is the homothety ofe: Let ρ >[ρ]: Recall that the mass distribution μ is called regular if μ t! (dffi) Ω ρr ρ 1 dr (1:1:7) in D -topology as t!1: (dffi) is a measure on the unit circle which is necessarily positive. Let ρ be an integer number p = [ρ]: Then the mass distribution is called regular if, in addition to (1.1.7), ffi R (z;μ t ;p) converges in D -topology as t!1 for some R: Since ffi R (z;μ t ;p) is a homogeneous harmonic polynomial, the convergence in D -topology is equivalent to uniform convergence in every bounded domain. Journal of Mathematical Physics, Analysis, Geometry, 27, vol. 3, No. 1 7

4 Vladimir Azarin In such terms Levin Pfluger's theorem (see [9, Chs. 2, 3], [1, Ch. 3, Th. 3]) may be formulated as follows. Levin Pfluger's Theorem. If u is a CRG-function, then its mass distribution is regular and vice versa Let ^ρ = fρ > ρ 1 > ::: > ρ n > g be a finite monotonic system of numbers. We call a function u 2 SH(ρ) completely ^ρ-regular if u t = h ρ + t ρ 1 ρ h ρ1 + :::+ t ρn ρ h ρn + t ρn ρ o(1); (1:2:1) where h ρ is a ρ-t.c. function and h ρj (z); j =1; 2;:::;n, are of the form of (1.1.2) with the corresponding h's being the differences of ρ j -t.c. functions. Therefore h ρj can be represented in the form of (1.1.4) or (1.1.6) with 's being the functions of bounded variation. Besides, o(1)! in D topology. Let ρ>[ρ] and ρ j 2 ([ρ];ρ), j =1; 2;:::;n. We callμ 2M(ρ) ^ρ-regular if as t!1,where μ t = μ (ρ) + j=n X j=1 t ρ j ρ μ (ρj ) + t ρn ρ o(1)) (1:2:2) μ (ρ) = ρ(dψ) Ω ρr ρ 1 dr; (1:2:3) with ρ positive andsummable, and μ (ρj ), j =; 1;:::;n, are of the same form as ρ = ρ j, j =; 1;:::;n, and arbitrary (ρj )'s that are the functions of bounded variation on the circle. If o(1)! in D -topology, then μ is ^ρ-regular in D -topology. However it is possible to say that μ is ^ρ-regular in other topology if o(1)! in this topology. Theorem Let ρ>[ρ] and [ρ n ;ρ] N = ;: If u is completely ^ρ-regular in D -topology then its mass distribution μ is ^ρ-regular in D topology. ^ρ-regular in C 1 p;p+1 *-topology, then u is completely ^ρ-regular in D -topology. If μ is Let us notice that the classical Levin Pfluger theorem of completely regular growth function for noninteger ρ can be obtained from here by using the following Proposition Let μ 2 M(ρ) and μ t! μ (ρ) in D as t! 1: Then the same holds in C 1 p;p+1λ. We suppose further that ρ is an exponent of the convergence of μ. Let us consider the situation, when ^ρ consists of noninteger numbers, but the interval (;ρ) contains integer numbers. Theorem Let u t have the representation u t = h ρ + t ρ 1 ρ h ρ1 + :::+ t ρn ρ h ρn + [ρ] X 1 <fa k z k gt k ρ + t ρn ρ o(1); (1:2:4) 8 Journal of Mathematical Physics, Analysis, Geometry, 27, vol. 3, No. 1

5 On the Polynomial Asymptotics of Subharmonic Functions of Finite Order where o(1)! in D : Then μ t = μ (ρ) + j=n X j=1 with o(1)! in D : The inverse theorem is the following t ρ j ρ μ (ρj ) + t ρn ρ o(1) (1:2:5) Theorem Let u 2 SH(ρ) and its mass distribution have the representation (1.2.5) with o(1)! in C 1 p;p+1 * and 2ß for all k, ρ>k>ρ j. Then (1.2.4) holds for u t with o(1)! in D : e ikffi ρj (dffi) = (1:2:6) Let us notice that the conditions (1.2.6) are not necessary for the validness of (1.2.4). The similar theorems can be formulated for the case when ρ or some of ρ j are integers. I am grateful to Prof. V. Logvinenko for his valuable notes. 2. Proofs 2.1. Consider the case when ρ>[ρ] and [ρ n ;ρ] N = ;. Let u t have the representation (1.2.1) and the remainder term be o(1) in D -topology. Applying to (1.2.1), the operator (1=2ß) (here is the Laplace operator) we obtain (1.2.2), as (1=2ß) u t = μ t ; (1=2ß) h ρj = ρj (dffi), j =;:::;n, and (1=2ß) o(1) = o(1) since the Laplace operator is continuous in D -topology. The first assertion of Th is proved. Let (1.2.2) hold with o(1) in C 1 *. Apply to it the operator p;p+1 AdΛ ρ which is conjugated to Ad ρ [ffl] := C n H(z= ; [ρ]) ffl (dxdy) that acts from D to C 1 p; p +1: By definition, for g 2Dwe have <Ad Λ ρμ t ;g >=< μ t ;Ad ρ [g] >: Now substitute (1.2.2) for μ t. The integral of the first n terms of (1.2.2 ) are, in fact, the first n terms of (1.2.1). Let us verify it. Journal of Mathematical Physics, Analysis, Geometry, 27, vol. 3, No. 1 9

6 Vladimir Azarin We have <μ (ρj );Ad ρ [g] > z = g(z)dxdy H(z=re iψ ;p) j(dψ)ρ j r ρ j 1 dr: Counting the inner integral on dr (see, [11, Ch. 1, 17, footnote 21]), we obtain 1 H(z=re iψ ;p)ρ j r ρ j 1 dr = Hence, using (1.1.4), we obtain 1 2ρ j sin ßρ j Λ cos ρ(arg z ψ ß)jzj ρj : (2:1:1) <μ ρj ;Ad ρ [g] > z =<h ρj :g > : (2:1:2) The last term is t ρn ρ o(1) where o(1) is understood in C 1 p;p+1 *. The function Ad ρ [g] is a canonical potential of the function g 2 D: Thus Ad ρ [g] 2 C 1. p;p+1 Therefore < o(1);ad ρ [g] > z! as t! 1: This proves the second assertion of Th Let us prove Proposition P r o o f. Let g 2 C 1 p;p+1 : Let fi 1, fi 2, fi 3 be a partition of unity by infinitely differentiable functions, such that supp fi 1 ρ (;ffl), supp fi 2 ρ (ffl=2; 2R), supp fi 3 ρ (R; 1): Then g(z)μ t (dxdy) =I 1 (t) +I 2 (t) +I 3 (t); where C I j (t) = g(z)fi j (jzj)μ t (dxdy); j =1; 2; 3: The first integral has the estimate C ji 1 (t)j»lim ffi! ffl ffi Cr p μ t (dr); because g is O(jzj p ) as z! : Integrating by parts, we obtain I 1 (t)» C 2 4 μt (ffl)ffl ρ p + lim ffi! ffl ffi 3 r p 1 μ t (r)(dr) 5 : 1 Journal of Mathematical Physics, Analysis, Geometry, 27, vol. 3, No. 1

7 On the Polynomial Asymptotics of Subharmonic Functions of Finite Order Since μ(r)» Cr ρ, also μ t (r)» Cr ρ : Thus uniformly with respect to t: In the same way we obtain uniformly with respect to t. Since μ t! μ ρ in D and gfi 2 2D,wehave I 2 (t)! Moreover, (2.2.1),(2.2.2), and (2.2.3) imply that C I 1 (t)» Cffl ρ p (2:2:1) I 3 (t)» CR ρ p 1 (2:2:2) g(z)fi 2 (jzj)μ ρ (dxdy); t!1: (2:2:3) <g;μ t >!< g;μ ( ρ) > for every g 2 C 1 p;p+1 because ffl can be chosen to be arbitrarily small and R can be selected to be arbitrarily large. For proving Th we should only repeat the first part of the proof of Th P r o o f o f T h e o r e m As in the proof of Th we apply the operator Ad Λ ρ to μ t and evaluate <μ ρj ;Ad ρ [g] > z. Because of (1.2.3), where <μ (ρj );Ad ρ [g] > z =<ρ j r ρ j 1 ;< ρj ;Ad ρ [g] > ffi > r ; 2ß < ρj ;Ad ρ [g] > ffi := Ad ρ [g](re iffi ) ρj (dffi): Changing the order of integration and using (1.2.6) and (2.1.1), we obtain <μ (ρj );Ad ρ [g] > z =<μ (ρj );Ad ρj [g] > z =<h ρj ;g >: As it was explained in the proof of Th , <o(1);ad ρ [g] >! : Thus Ad Λ ρμ t = h ρ + t ρ1 ρ h ρ1 + ::: + t ρn ρ h ρn + o(1)t ρn ρ : (2:3:1) By Adamar's theorem (see, e.g., [12, Ch. 4.2]) X u(z) Ad Λ ρ μ(z) = [ρ] Thus (2.3.1) and (2.3.2) imply (1.2.4). k= <fa k z k g: (2:3:2) Journal of Mathematical Physics, Analysis, Geometry, 27, vol. 3, No. 1 11

8 Vladimir Azarin References [1] V.N. Logvinenko, On Entire Functions with eros on the Halfline. I. μ Teor. Funkts., Funkts. Anal. i Prilozh. 16 (1972), (Russian) [2] V.N. Logvinenko, Two Term Asymptotics of a Class of Entire Functions. μ Dokl. Akad. Nauk USSR 25 (1972), (Russian) [3] V.N. Logvinenko, On Entire Functions with eros on the Halfline. II. μ Teor. Funkts., Funkts. Anal. i Prilozh. 17 (1973), (Russian) [4] P.. Agranovich and V.N. Logvinenko, Analog of the Valiron Titchmarsh Theorem for Two-Term Asymptotics of Subharmonic Functions with Masses on a Finite System of Rays. μ Sib. Mat. h. 26 (1985), No. 5, (Russian) [5] P.. Agranovich and V.N. Logvinenko, Polynomial Asymptotic Representation of Subharmonic Function in the Plane. μ Sib. Mat. h. 32 (1991), No. 1, (Russian) [6] P.. Agranovich and V.N. Logvinenko, Exceptional Sets for Entire Functions. μ Mat. Stud. 13 (2), No. 2, [7] P.. Agranovich, On a Sharpness of Multiterm Asymptotics of Subharmonic Functions with Masses in a Parabola. μ Mat. fiz., anal., geom. 11 (24), (Russian) [8] P.. Agranovich, Massiveness of Exeptional Sets Multi-Term Asymptotic Representations of Subharmonic Functions in the Plane. μ J. Math. Phys., Anal., Geom. 2 (26), [9] V.S. Azarin, On the Asymptotic Behavior of Subharmonic and Entire Functions. μ Mat. Sb. 18 (1979), No. 2, (Russian) [1] A.A. Gol'dberg, B.Ya. Levin, and I.V. Ostrovskii, Entire and Meromorphic Functions. μ Ecyci. Math. Sci. 85 (1997), [11] B.Ya. Levin, Distribution of eros of Entire Functions. AMS, Providence, RI, 198. [12] W.K Hayman and B.P. Kennedy, Subharmonic Functions. Vol. I Acad. Press, London, New York, San Francisco, Journal of Mathematical Physics, Analysis, Geometry, 27, vol. 3, No. 1