CC BY
28Trudy Petrozavodskogo gosudarstvennogo universiteta
Seria “Matematika” Vypusk 17, 2010
UDK 517
ON ZEROS OF SOME ANALYTIC SPACES OF AREA NEVANLINNA TYPE IN HALFPLANE
R. F. Shamoyan, O. R. MihiC
We introduce two new scales of spaces of Nevanlinna type in a halfplane C+ and provide descriptions of their zero sets and based on it we present parametric representations of these classes.
§ 1. Introduction
One of the most important directions in the theory of holomorphic functions in halfplane is the study of various problems connected with zero sets of functions of these classes (see [1], [2]). In this note we introduce new analytic classes in halfplane and provide descriptions of their zero sets in C+, where C+ is a upper halfplane in C. We plan to provide some applications we presented in this note in a separate paper. The basic facts of the theory of analytic classes in the halfplane were proved by V. Krylov (see [5]) and E. Hille - Tamarkin independently (see also [4], [6]).
The goal of this paper is to find complete analogues of our recent results on zero sets from [7] for the case of spaces in the unit disk in a case of halfplane C+.
Let, as usual, H(C+) is a space of all analytic functions in upper halfplane.
Let T(f,T) (see [2]) be a Nevanlinna characteristic of f and for 0 < p < to, a > -1, let also
p OO /* oo
(C+) = {f € H(C+) : ya ln+ |f(x + iy)ldxdy < to}.
J 0 J — o
We introduce new spaces of analytic functions in C+.
(Np)i = {f € H(C+) :
© R. F. Shamoyan, O. R. Mihic, 2010
fO / rV ro \P
/ [ ln+ If (x + iy)|dxdy) yady < to; }
Jo \J 0 J — O )
(N%)2 = {f € H(C+) :
sup f f f ln+ If (x + iy)|dx J ya dyj y? < to}, ft > 0.
Cy<o Jo VJ— o J J
0<y<o Jo
We note that classes we introduced are Banach spaces for p > 1, a > —1 and complete metric spaces for p < 1.
Let further also n(y) = n(y, {zk}) = {card zk : Imzk > y}, {zk}O € C+. Let
/O
|F(x + iy)|pdx < C}, 0 < p < to.
-oo
The following classical result gives complete parametric representation of Hp - Hardy classes (see [4]).
Theorem 1. Let F € Hp(C+). Then F can be represented as F(z) = eiYIF(z)QF(z), z € C+, y € R,
where
If(z) = fiei“‘ f-fexp (1 S—o (V—t + <2+71) da«>)e“z,
i _ zk
F(zk) = 0, eiak-------— > 0, a is singular measure, a < 0,
i — zk
!O lda«! < to,
J-o 1+ t2 '
and
Qf(z) = exp (n ISz—t + p+r)log|F(t)|dt).
The goal of this note to use known Blaschke type products in a halfplane (see [2]).
b ( f t ) IT \ (21mzh t°dt \
Ba (z, {zk })= n expj ^o (t + i(zk — z))l+a ),
where z, {zk }o € C+, a > — 1,
^(Imzk)
1+“ < 00,
k0
for the study of zero sets of classes we introduced above. We note also the descriptions of zero sets provide also so called parametric representation of mentioned spaces. We formulate at the end of this section a known result on zero sets for N^ classes. By Z(f) we denote the zero set of an analytic function f in H(C+).
Theorem 2. (see [2]). Let a > —1. Then N^ (C+) class coincides with all functions f,f € H(C+) so that
/11 rmiw Ia
log |f (w)|—;----- dm2(w)},
:+ (i(z — w))2+“
(1 + a)21+a
where Aa =----------------- and also if Z(f) = {ak} then
n
^(Imak)2+a < to. (1)
k>0
Moreover condition (1) characterizes zero sets of N^ (C+) classes. Throughout the paper, we write C (sometimes with indexes) to denote a positive constant, which might be different at each occurrence (even in a chain of inequalities) but is independent of the functions or variables being discussed.
§ 2. Main results
In this section we provide main results of this note and give some preliminaries that were used by us during the proof of our results.
We note that as direct corollaries of our main theorems we also formulate parametric representation of (N?)1 and (Np ?)2 spaces in a halfplane. Various applications of such parametric representations for classes in a halfplane are well known (see [2]). Complete analogues of our parametric representations in case of classical Hardy classes in halfplane can be found
in [4L [6].
Theorem 1. Let 0 < p < to, ft > —1. The following is true: If {zk}o=1 is a zero set of a function f, f = 0, f € (Np )1, and lim ((t ln |f (t)|) < to.
? t—>OD
Then
Enk
2k(3+2p+1) < nk
If
2k(3+2p+1) " ,t,K n \ 2k
k=1
oo
V ______'Jk____ < oo
2k(3+2p+1) <
k=1
then BY(z, {zk}) for 7 > belongs to (N3)1 and converges uniformly within C+, 0 < p < to, ft > — 1.
Theorem 2 is an analogue of Theorem 1 for (Np 3)2 classes.
Theorem 2. Let 0 < p < to, a > —1, ft > 0. The following are equivalent:
(1) |zk}^=1 is a zero set of a function f, f = 0, f e (Np ^)2;
C
(2) n(y) ^ ya+3+P+1.
Moreover, if t > a + ft + p then Bt(z, {zk}) converges uniformly within C+ and belongs (Np^)2.
In following Theorem 3 and Theorem 4 we provide complete parametric representations of and (^NP 3) classes based on descriptions of
zero sets we provided above for these classes in Theorem 1 and Theorem 2. We have as immediate corollaries of Theorem 1 and Theorem 2.
Theorem 3. Let 0 <p< to, ft > —1, lim ((t ln |f (it)|) < to, and f e
t——+ TO
(N3)1, f is a non zero function. Then (N3)1 coincides with a class of the functions that admits representation f (z) = BY(z, {zk})(exph(z)), z e C+ where {zk} is an arbitrary sequence from C+ that satisfies
TO p
En% <
2k(^+2p+1) <
k=1
ft +1
7 > ----- and h e H(C+), moreover
p
ry /■ to \ p
/ / |h(x + iy)|dxdy) y3dy < to, ft> —1.
Theorem 4. Let 0 < p < to, a > —1, ft > 0, 7 > a + ft + p, f = 0, f e (Np 3)2. Then (Np 3)2 coincides with a class of functions that
1
admits representation f (z) = BY(z, {zk})(exp h(z)), z G C+, where {zk}
C
is an arbitrary sequence from C+ that satisfies n(y) < a+p+p+1 and h G H(C+), moreover
r y / r TO _
sup y^ i |h(x + iyj)\dx\ yady < to.
0<y<TO J0 V J— to )
Remark. Let w > 0 on (0, to) and w G L1(0, A) for every A > 0. Let assume for every A > 0 there are numbers mu, Mu, , so that mu, G
(0,1), Mu > 0 and so that mu < (.^ < Mu, X G [qw, 1], y > 0. We
^(y)
finally note that results can be extended to w(y) more general weights so that ya replaced by general function w(y). We among other things for the proofs use the following lemma which is interesting by itself.
Lemma 1. Let {zn}TO be a sequence from C+ for which
OO
^(Jmzn)1+^ < to
n=1
and for some constant C > 0 and a > 1 n(y, {zk}) < Cy—a, y > 0. Then for p > 1, f3 > a — 1,
/TO
| ln Bp(x + iy, {zk})|pdx < K(a,ft,p)Cpy1—ap,
-OO
for some constant K.
Bibliography
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[2] Djrbashian A. M. Functions of a bounded type in the halfplane. New-York: Springer, 2005.
[3] Hille E., Tamarkin J. On the absolute integrability of Fourier transform // Fund. Math. 25. 1935. P. 329-352.
[4] Koosis P. Introduction to Hp spaces. lms. note series 40. Cambridge: Cambridge University Press, 1980.
[5] Krilov V. On functions regular in the halfplane // Russian Mat. Sbornik. V. 6 (48). 1939. P. 95-138.
[6] Nikolski N. Operators, functions and systems / Mathematical surveys and monographs. V. 92. 2002.
[7] Shamoyan R., Li H. Descriptions of zero sets and parametric representations of certain analytic area Nevanlinna type classes in the unit disk // Bulletin Georgian Math. Society. (In Press)
Bryansk University, Bryansk, Russia.
E-mail: [email protected] Fakultet organizacionih nauka Jove Ilica 154, Belgrade, Serbia.
E-mail: [email protected]
