IN THE UNIT DISK
IGOR CHYZHYKOV and YURII LYUBARSKII
We study approximation of subharmonic functions in the unit disc by logarithms of moduli of analytic functions in the uniform metric. We generalize known results in this direction, in particular, we get rid of growth restrictions on the Riesz measure of a subharmonic function.
AMS 2010 Subject Classification: Primary 31A05, Secondary 30J99.
Key words: subharmonic function, uniform approximation, analytic function, Riesz measure.
1. INTRODUCTION
This article studies approximation of subharmonic functions in domains of the complex plane by logarithms of moduli of analytic functions. Such approximation is now a widely used tool in analysis and (in various modifica- tions) has been considered in a number of works, we refer the reader to [3, 21]
to mention just a few.
The cases when the Riesz measure of subharmonic function is located on a system of curves have been treated in [1, 15, 16] (see also Section 10.5 in [13]).
In the more general situation, the following two cases are of the most interest:
a. Subharmonic functions which are defined in the whole complex plane;
we approximate them by logarithms of moduli of entire functions;
b. Subharmonic functions in the unit disk, respectively bounded domain.
These cases are closely related: the techniques developed initially for approximation in the whole plane have their counterparts for the disk case.
The construction of the approximating analytic function is carried out via the “atomization procedure”: Let µbe the Riesz measure of the subharmonic function u. We split µ into pieces, each carrying an integer load and then replace each such piece by the corresponding number of single unit atoms.
These atoms are located at zeros of the approximating analytic function.
There are two major obstacles in performing the atomization procedure:
– decomposition of the Riesz measure into “integer” pieces may be very non-trivial especially in the case of highly irregularµ; in the recent works, this can be done by applying the Yulmukhametov partition lemma, see [25];
MATH. REPORTS15(65),4(2013), 359–371
– after the desired decomposition is found, one has to estimate the appro- ximation error; this is related to precise and sometimes very delicate estimates.
In the present note, we concentrate on the second obstacle for the disk case: assuming that (rather regular) partition is already found we give a precise estimate of the approximation error, one of the purposes of the present note is to trace how the techniques developed for entire functions can be adjusted for this (perhaps more complicated) case.
We begin with a brief overview of the results already known. In the full generality the problem has been posed in [2]. The author considered classes functionsusubharmonic in the whole complex plane Cand having finite order ρ,i.e. such that, for each ε >0
u(z)≺(1 +|z|)ρ+ε, z∈C, and proved that there exist an entire function f satisfying
u(z)−log|f(z)|=o(|z|ρ), z→ ∞, z 6∈E for an appropriate exceptional set E.
A breakthrough was achieved in [11, 25], which obtain logarithmic esti- mates in uniform and integral norms, respectively. We denote by SH(C) the set of all functions subharmonic in the whole complex plane. Given E ⊂ C and t >0 letU(E, t) ={ζ ∈C: dist(ζ, E)< t}.
Theorem A ([25]). For any functionu∈SH(C) of orderρ∈(0,+∞), and α > ρ there exist an entire function f and a setEα ⊂C such that
(1.1)
u(z)−log|f(z)|
≤C(α) log|z|, z→ ∞, z 6∈Eα, and Eα can be covered by a family of disksU(zj, tj), j∈N, withP
|zj|>Rtj = O(Rρ−α), (R→+∞).
In order to formulate the estimate in the integral norm, we need additional notation. Givenu∈SH(C) we denote
Q(r, u) =
(logr, uis of finite order,
logr+ logn(r, u), r→+∞, uis of infinite order, wheren(r, u) =µ(U(0, r)).
Theorem B ([11]). For an arbitrary u∈SH(C) and p >0 there exists an entire function f such that
u(reiθ)−log|f(reiθ)|
p =O(Q(r, u)), r → ∞, r6∈E.
The exceptional set E is empty in the case of functions of finite order and can be chosen of finite linear measure for functions of infinite order.
Theorem C ([18]). Let u∈SH(C). Then, for eachq >1/2, there exist R0>0 and an entire function f such that
(1.2) 1
πR2 Z
|z|<R
u(z)−log|f(z)|
dm(z)< qlogR, R > R0.
The constant 1/2 in this theorem is optimal: the functionu(z) = 1/2 log|z|
cannot be approximated with anyq <1/2. This is true even if the total Riesz massµofuis infinite: in the case when suppµhas sufficiently big gaps so that a point mass of size 1/2 were sufficiently isolated, one cannot achieve better approximation. On the other hand, if suppµ does not have Hadamard gaps, i.e. if there existsa >1 such that
µ({z:R <|z| ≤aR})>1, R > R1 one can replace (1.2), by
(1.3) 1
πR2 Z
|z|<R
u(z)−log|f(z)|
dm(z) =O(1), R → ∞.
see [18].
This result was further developed in [6]; in this article the size of possible gaps was precisely related to the approximation error.
Let Φ be the class of slowly growing functions ψ: [1,+∞)→(1,+∞) (in particular,ψ(2r)∼ψ(r) as r →+∞).
Theorem D ([6]). Let u ∈ SH(C), µ = µu. If for some ψ ∈ Φ there exists a constant R1 satisfying the condition
(1.4) (∀R > R1) :µ({z:R <|z| ≤Rψ(R)})>1, then there exists an entire function f such that (R≥R1)
1 R2
Z
|z|<R
u(z)−log|f(z)|
dm(z) =O(logψ(R)).
(1.5)
Examples from [6] show that estimate (1.5) is sharp in the class of sub- harmonic functions satisfying (1.4).
The structure of the exceptional set is a subject of special interest. In particular, it is important to reduce the exceptional set to a neighborhood of the zero set of the approximating entire function. This is impossible; in general, one has to impose additional restriction on the regularity of the Riesz measure.
One of possible ways to do can be found in Theorem 3 [18] where notions of locally regular measure and partition of slow variation were introduced. A
disc counterpart can be found in [7]. Another way see in [22] (for the disk case) and in [20] (for the plane) demands the local doubling condition. In these articles, the authors apply another approach to atomization based on quadrature formulas and in particular, obtain analog of (1.3) with precise control over the exceptional set.
When considering bounded domains we are interested mainly in uniform estimates. We refer the reader to the problem posed by M. Sodin (Question 2 in [24, p. 315]):
Given a Borel measure µwe define its logarithmic potential as Uµ(z) =
Z
log|z−ζ|dµ(ζ).
Question. Let µ be a probability measure supported by the square Q= {z=x+iy:|x| ≤ 12,|y| ≤ 12}. Is it possible to find a sequence of polynomials Pn, degPn=n, such that
Z Z
|x|≤1
|y|≤1
|nUµ(z)−log|Pn(z)||dxdy=O(1), n→ ∞
An affirmative answer to this question can be found in [18]. In [7] it is proved that approximation rate is actually uniformly bounded.
Proposition ([7]). Let µ be a measure supported by the square Q, and µ(Q) = N ∈ N. Then there is an absolute constant C and a polynomial PN
such that
Z Z
Ξ
|Uµ(z)−log|PN(z)||dxdy < C, where Ξ ={z=x+iy:|x| ≤1,|y| ≤1}.
A rough analysis of the proof gives the following upper estimate for the constant C <3·104. The proposition allows to prove the following theorem which can be considered as a counterpart of Theorems C and D for subharmonic functions in the unit disk.
Theorem E ([7]). Let u ∈SH(D). There exist an absolute constant C and an analytic function f in D such that
Z
D
u(z)−log|f(z)|
dm(z)< C.
Theorem E does not provide the
u(z)−log|f(z)|=O(1), z∈D\E
for any “small” set E. As in the case of entire functions, we need to im- pose some regularity conditions on the Riesz measure. For the disk case, such conditions has been introduced in [7]. Unfortunately, the results of this work cannot be applied to functions of slow growth, namely, if B(r, u) = O(log1−r1 (log+log1−r1 )α), r ↑ 1, α ≥ 0, and also, to those growing faster than power functions,e.g. exp
1 1−|z|
β
,β >0. Here, we improve this result.
Some applications of results of such type to constructing of analytic func- tions with prescribed asymptotic values can be found in ([1], [13] Chapter 10) behavior of the minimum modulus in [5], to value distribution theory in [8, 9], and to interpolation problems [4, 20, 23]. Actually, in [4] a general result on ap- proximation of unbounded functions of the formh(z) =h(|z|) that grow faster than polynomials do by log|f|, where f is an analytic function, is obtained.
We also refer the reader to [20, 22] which apply the atomization techniques to solution of the ¯∂-problem in weighted spaces. We consider the general problems of this type for the case of the unit disc.
2. UNIFORM APPROXIMATION
In this section, we prove some counterparts of the results from [7, 18].
We start of with notions, which reflect regularity properties of measures.
Definition 1.We say that a function b : [0,1)→ (0,∞) is a function of regular variation if, for all r <1 one has r+b(r)<1 and
a)
(2.1) b(r+b(r))b(r);
b)
(2.2) (∀c >0) : sup
0<r<1
Z 3
4(1−r) cb(r)
sup|ρ−r|≤τ|b0(ρ)|
τ dτ <+∞.
Remark. It follows from the definition thatb(r)→0 asr %1. As a rule, b is chosen to be nonincreasing in applications.
By rectangle Q ⊂ D we mean a set of the form Q = {z = reiθ;r1 <
r < r2, θ1 < θ < θ2}. The sides of Q have the length λr(Q) = r2−r1 and λθ(Q) =θ2−θ1, respectively.
Definition 2. We say that a measure µ in D admits a regular partition with respect to a function b of regular variation if there exist sequences (Q(l)) of rectangles inD and decomposition ofµ:
µ=X µ(l)
such that
i) suppµ(l)⊂Q(l),µ(l)(Q(l)) = 2;
ii) for some N each z∈Dbelongs to at most N variousQ(l)’s;
iii) For somec >0 we have
0< c−1< λθ(Q(l))/λr(Q(l))< c <∞;
iv) dl:= diamQ(l)b(dist(Q(l),0)).
Definition 3. Given a function b satisfying assumptions of Definition 1, we say that a measureµ is locally regular with respect to (w.r.t.) b if
Z b(|z|) 0
µ(U(z, t))
t dt=O(1), r0 <|z|<1, for some constant r0∈(0,1).
Let Zf stand for the zero set of analytic functionf.
Theorem 1. Let u ∈ SH(D), b: [0,1) → (0,+∞) satisfy conditions of Definition 1. Let µu admit a partition of regular variation, and µu be locally regular w.r.t. b. Then there exists an analytic functionf inDsuch that∀ε >0
∃r1∈(0,1)
(2.3) log|f(z)| −u(z) =O(1), r1 <|z|<1, z6∈Eε,
where Eε ={z∈D: dist(z, Zf)≤εb(|z|)}, and for some constant C >0 (2.4) log|f(z)| −u(z)< C, z∈D.
Moreover, Zf ⊂S
ζ∈suppµU(ζ, b(|ζ|)) for some K >0.
Remark. In [7] the theorem was proved under the additional restrictions (2.5)
Z 1 0
bp−1(t)
(1−t)pdt <+∞ for some p≥2,
andb(r1)b(r2) when 1−r11−r2. We release restriction (2.5) by applying some ideas and techniques from [23], and also relax slightly the second one (see Remarks at the end of the section).
Examples. Theorem 1 can be applied to subharmonic functions of the form u = logα1−|z|1 , α ≥ 1, u(z) = (1−|z|)1 β, β > 0. Indeed, we can choose
b(r) 1−r
(log1−r1 )α−12
, and b(r) (1−r)1+β2, respectively. Note that the case α= 1 is covered by [23]. It is also possible to take b(r) 1−r
(log+log1−r1 )γ,γ ≥0 which does not satisfy (2.5). All of these b(r) satisfy (2.1) and (2.2). On the
other hand, b(r) = 1/expn{1−r1 }, where expn denotes the nth iteration of the exponent,n∈N, satisfies Definition 1 as well.
Proof of Theorem 1. LetQ={Q(l)}be a partition of slow variation with respect tobandµ=P
µ(l) be the corresponding decomposition ofµ. Letζl(1), ζl(2),l= 1,2, . . .be the solutions of the system
(2.6)
ζl(1)+ζl(2)= Z
Q(l)
ζdµ(l)(ζ),
(ζl(1))2+ (ζl(2))2 = Z
Q(l)
ζ2dµ(l)(ζ).
Consider the functions Vl(z)def=
Z
Q(l)
log
z−ζ 1−zζ¯
− 1
2log
z−ζl(1) 1−zζ¯l(1)
− 1
2log
z−ζl(2) 1−zζ¯l(2)
dµ(l)(ζ), (2.7)
V(z)def= X
l
Vl(z).
(2.8)
LetZ={ζl(1)}l∪{ζl(2)}l. Givenε >0 denoteEε={z∈D: dist (Z, z) <
εb(|z|)}, this is our exceptional set.
Lemma 1. For each ε > 0 the series (2.8) converges uniformly on each compact set in D\Eε and, for some constant C=C(ε)>0,
|V(z)|< C, z6∈Eε
(2.9)
−V(z)< C, for all z∈D (2.10)
Assume Lemma 1 was already proved. Then the statement of Theorem 1 follows readily, indeed U(z) :=u(z)−V(z) is a subharmonic function with the Riesz measure
µU =X
l
δζl(1)+X
l
δζl(2).
It follows that U(z) = log|f(z)| for some f ∈ Hol (D). This is the desired function for relations (2.3) and (2.4) are just reformulations of (2.9) and (2.10).
Proof of Lemma 1. For a partitionQ, let ζl
def= 1 2
Z
Q(l)
ζdµ(l)(ζ), dl= diamQ(l).
It follows from (2.6) that (see e.g. [6, 18])
|ζl(j)−ζl| ≤dl, j∈ {1,2}.
(2.11)
Fix z∈Dand split the partition as Q=Q+∪ Q−, whereQ−={Q(l)∈ Q: dist(z, Q(l))> 1−|z|4 };Q+=Q \ Q−.
We then have V(z) =
X
Q(l)∈Q−
+ X
Q(l)∈Q+
Vl(z) =V−(z) +V−(z).
Each term in the right-hand side will be estimated separately.
Estimate of V−(z).
We observe that there existsc >1 such that for each Q(l)∈ Q−
(2.12) 1
c ≤ |1−zζ¯ 0|
|1−zζ¯ 00| ≤c, ζ0, ζ00∈Q(l) because
|1−zζ¯ 0|
|1−zζ¯ 00| |1−zζ¯ 0|
|z−ζ0|
|z−ζ00|
|1−zζ¯ 00| and both factors are bounded from above and from below.
Now, for Q(l) ∈ Q− take a branch L(ζ) = log 1−¯ζ−zzζ
in Q(l). We then have
Vl(z) = Re Z
Q(l)
L(ζ)−1
2 L(ζl(2)) +L(ζl(1))
dµ(l)(ζ)
We apply the order two Taylor formula with the integral remainder term:
L(ζ)−L(ζl(1)) =L0(ζl(1))(ζ−ζl(1)) +
ζ
Z
ζl(1)
L00(s)(ζ−s) ds.
An explicit calculation shows
L00(ζ) = (1− |z|2)(¯z(ζ−z)−(1−ζz))¯ (1−ζ¯z)2(ζ−z)2 , and hence,
(2.13) |L00(ζ)| ≤ 2(1− |z|2)
|1−ζz||ζ¯ −z|2 ≤ 2c2(1− |z|2)
|1−ζz|¯3 , ζ ∈Q(l). Now,
|Vl(z)|= Re
Z
Q(l)
L(ζ)−L(ζl(1))−1
2(L(ζl(2))−L(ζl(1))
dµ(l)(ζ) =
= Re
Z
Q(l)
L0(ζl(1)) ζ−1
2(ζl(1)+ζl(2)) +
+ Z ζ
ζl(1)
L00(s)(ζ−s)ds−1 2
Z ζ(2)
l
ζl(1)
L00(s)(ζl(2)−s)ds
dµ(l)(ζ) =
= Re
Z
Q(l)
Z ζ ζ(1)l
L00(s)(ζ−s)ds−1 2
Z ζl(2) ζ(1)l
L00(s)(ζl(2)−s)ds
dµ(l)(ζ) , (2.14)
and (we remind thatdl= diamQ(l)) by (2.13), (2.11)
Z ζ ζl(1)
L00(s)(ζ−s) ds
≤2c2(1− |z|2) Z
[ζl(1),ζ]
|ζ−s||ds|
|1−ζz|¯3 ≤
≤ 2c2(1− |z|2)|ζl(1)−ζ|2
|1−ζz|¯3 ≤ 8c2d2l(1− |z|2)
|1−ζz|¯3 ,
Z ζ(2)l ζl(1)
L00(s)(ζ−s) ds
≤ 8c2d2l(1− |z|2)
|1−ζl(2)z|¯3 . Finally,
|Vl(z)| ≤ Z
Q(l)
8c2(1− |z|2)d2l 1
|1−ζz|¯3 + 1 2|1−ζl(2)z|¯3
dµ(l)(ζ)≤
≤12c2(1− |z|2)d2l max
ζ∈U(ζl,dl)
1
|1−ζz|¯3. Relation (2.12) and properties iv) and iii) now yield
|V−(z)| ≤ X
Q(l)∈Q−
|Vl(z)| ≤12c2 X
Q(l)∈Q−
d2l max
ζ∈U(ζl,dl)
1− |z|2
|1−ζz|¯3 ≤
≤C(1− |z|2) X
Q(l)∈Q−
Z
Q(l)
dm(ζ)
|1−zζ¯ |3 ≤
≤CN(1− |z|2) Z
D
dm(ζ)
|1−zζ¯ |3 ≤C1. (2.15)
This completes the estimate of V−(z).
In order to estimate the term V+, which represents the contribution of rectangles located “close” to the point z, we need the third order Taylor ex- pansion:
L(ζ)−L(ζl(1)) =L0(ζl(1))(ζ−ζl(1))+1
2L00(ζl(1))(ζ−ζl(1))2+1 2
ζ
Z
ζl(1)
L000(s)(ζ−s)2ds,
where the function L(ζ) = log 1−¯ζ−zzζ
is still well-defined in any Q(l),z6∈Q(l). We have
L000(ζ) = 2
(ζ−z)3+ 2
(1z¯−ζ)3, and |L000(ζ)| ≤ 4
|ζ−z|3, ζ ∈Q(l), z∈D\Q(l). Now,
|Vl(z)|= Re
Z
Q(l)
L0(ζl(1)) ζ−1
2(ζl(1)+ζl(2)) +
+L00(ζl(1)) 2
ζ2−(ζl(1))2+ (ζl(2))2
2 +ζl(1)(ζl(1)+ζl(2)−2ζ)
+ +1
2 Z ζ
ζl(1)
L000(s)(ζ−s)2ds− 1 4
Z ζl(2) ζl(1)
L000(s)(ζl(2)−s)2ds
dµ(l)(ζ) =
=
Re Z
Q(l)
1 2
ζ
Z
ζl(1)
L000(s)(ζ−s)2ds−1 4
ζ(2)l
Z
ζl(1)
L000(s)(ζ−s)2ds
dµ(l)(ζ)
≤6d3l max
s∈U(ζl,dl)
1
|s−z|3,
provided that dist(z, Q(l))≥3dl. Therefore, by the properties of the functionb
X
Q(l)∈Q+, dist(Q(l),z)>Cb(|z|)
Vl(z)
≤C2 X
Q(l)∈Q+, dist(Q(l),z)>Cb(|z|)
max
s∈U(ζl,dl)
b(|ζl|)d2l
|s−z|3 ≤
≤C3
Z
3
4(1−|z|)>|z−ζ|>Cb(|z|)
b(|ζ|) dm(ζ)
|z−ζ|3 ≤C5b(|z|)
Z
3
4(1−|z|)>|z−ζ|>Cb(|z|)
dm(ζ)
|z−ζ|3+ +C5
Z
3
4(1−|z|)>|z−ζ|>Cb(|z|)
|b(|ζ|)−b(|z|)|dm(ζ)
|z−ζ|3 ≤
≤C6b(|z|)
3 4(1−|z|
Z
Cb(|z|)
dτ τ2 +C6
3 4(1−|z|)
Z
Cb(|z|)
sup|ρ−|z||≤τ|b(ρ)−b(|z|)|dτ
τ2 ≤
≤C7+C6
3 4(1−|z|)
Z
Ccb(|z|)
sup|ρ−|z||≤τ|b0(ρ)|dτ
τ ≤C8.
(2.16)
So, it remains to estimate contributions only from those Q(l) which are located in a very close neighborhood of z i.e. dist(z, Q(l)) < Cb(|z|). There are just a finite number of suchQ(l)’s.
Let Q(l)∩U(z, Cb(|z|)6=∅. Forz6∈Eε we have log|z−ζl(k)|= logb(|z|) + log|z−ζl(k)|
b(|z|) = logb(|z|) +O(1), 1≤k≤2.
Therefore, Vl(z) =
Z
Q(l)
log|z−ζ| − 1 2
2
X
k=1
log|z−ζl(k)|
dµ(l)(ζ)−
−1 2
Z
Q(l)
log |1−zζ¯ |2
|1−zζ¯ l(1)||1−zζ¯ l(2)|dµ(l)(ζ) =:B1+B2.
We have|1−zζ¯ | 1−|z|2, that isζbelongs to a hyperbolic neighborhood of z. That is why B2 =O(1).
In order to estimateB1,we use local regularity ofµat the pointz, see (2).
Let as before µz(t) =µ(U(z, t)). We have J3 =
Z
Q(l)\U(z,b(|z|))
log|z−ζ|dµ(l)(ζ) + Z
U(z,b(|z|))
log|z−ζ|dµ(l)(ζ)−
−2 logb(|z|) +O(1) =µ(l)(Q(l)\U(z, b(|z|)) logb(|z|) +O(1)+
+ Z b(|z|)
0
logtdµ(l)z (t)−2 logb(|z|) =µ(l)(Q(l)\U(z, b(|z|)) logb(|z|)+
+O(1) +µ(l)(U(z, b(|z|)) logb(|z|)− Z b(|z|)
0
µ(l)z (t) t dt−
−2 logb(|z|) =− Z b(|z|)
0
µ(l)z (t)
t dt+O(1) =O(1) by the regularity ofµu w.r.tb(t). Together with (2.16) it yields
X
Q(l)∈Q+
|Vl(z)|=O(1), z6∈Eε. This completes the proof of the theorem.
We complete this section with some remarks.
Remarks.
1. As in [7], we can state that under the assumption of Theorem 1 we have T(r, u)−T(r,log|f|) = O(1), where T(r, u) = 2π1 R2π
0 u+(reiθ)dθ is the Nevanlinna characteristic of the subharmonic function u.
2. One can relax slightly the requirement on Q(l) to be rectangles by the condition (diamQ(l))2 ≤ cm(Q(l)), which trivially holds for our definition of rectangle.
3. In general, we need not assume the existence of the derivative ofb. In this case, one have to replace condition (2.2) by
(∀c >0) : sup
0<r<1
Z 1−r cb(r)
sup|ρ−r|≤τ|b(ρ)−b(r)|
τ2 dτ <+∞.
4. We performed atomization by choosing just two points in each rect- angle Q(l) and annihilating just two first moments (see (2.6)). As in [18], annihilating additional moments does not lead to improvement of approxima- tion. This is in striking contrast with techniques in [22] where the author uses annihilation of a bigger number of moments and then applied quadrature formulas.
Acknowledgments. This paper was inspired by the “Conference on Complex Analy- sis and Related Topics, the 13th Romanian-Finnish Seminar” (Ploie¸sti, Romania, June 26–30, 2012). The first author thanks the organizers for hospitality and financial sup- port. The second author is partially supported by the NFR project 213638 and also, by the research project at the Center of Advanced Study 2012/2013, Oslo.
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Received 17 February 2013 Faculty of Mechanics and Mathematics, University Universytets’ka 1,
Lviv Ivan Franko National, 79000, Lviv, Ukraine
and
Cardinal Stefan Wyszy´nski University in Warsaw, Faculty of Mathematics and Natural Studies,
W´oycickiego 1/3, PL-01-938, Warszawa, Poland chyzhykov@yahoo.com
Norwegian University of Science and Technology, Department of Mathematical Sciences,
7491, Trondheim, Norway yura@math.ntnu.no
