On the Growth of the Cauchy-Szegő Transform in the Unit Ball
The growth of analytic and harmonic functions in the unit ball Bn represented by the Cauchy-Stieltjes or Poisson-Stieltjes integral is studied. A description of the growth is given in terms of smoothness of the Stieltjes measure. Изучен рост аналитических и гармонических функций в единичном шаре, пр...
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Chyzhykov, I. Voitovych, M. 2017-05-28T18:46:07Z 2017-05-28T18:46:07Z 2015 On the Growth of the Cauchy-Szegő Transform in the Unit Ball / I. Chyzhykov, M. Voitovych // Журнал математической физики, анализа, геометрии. — 2015. — Т. 11, № 3. — С. 236-244. — Бібліогр.: 11 назв. — англ. 1812-9471 DOI: 10.15407/mag11.03.236 MSC2000: 32A26 (primary); 32A25 (secondary) https://nasplib.isofts.kiev.ua/handle/123456789/118150 The growth of analytic and harmonic functions in the unit ball Bn represented by the Cauchy-Stieltjes or Poisson-Stieltjes integral is studied. A description of the growth is given in terms of smoothness of the Stieltjes measure. Изучен рост аналитических и гармонических функций в единичном шаре, представленых интегралом Коши-Стилтьеса или Пуассона-Стилтьеса. Описание роста дается в терминах гладкости меры Стилтьеса. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Журнал математической физики, анализа, геометрии On the Growth of the Cauchy-Szegő Transform in the Unit Ball Article published earlier |
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The growth of analytic and harmonic functions in the unit ball Bn represented by the Cauchy-Stieltjes or Poisson-Stieltjes integral is studied. A description of the growth is given in terms of smoothness of the Stieltjes measure.
Изучен рост аналитических и гармонических функций в единичном шаре, представленых интегралом Коши-Стилтьеса или Пуассона-Стилтьеса. Описание роста дается в терминах гладкости меры Стилтьеса.
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https://nasplib.isofts.kiev.ua/handle/123456789/118150 |
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On the Growth of the Cauchy-Szegő Transform in the Unit Ball / I. Chyzhykov, M. Voitovych // Журнал математической физики, анализа, геометрии. — 2015. — Т. 11, № 3. — С. 236-244. — Бібліогр.: 11 назв. — англ. |
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2025-11-25T22:46:31Z |
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1850569977738821632 |
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Journal of Mathematical Physics, Analysis, Geometry
2015, vol. 11, No. 3, pp. 236–244
On the Growth of the Cauchy–Szegő Transform
in the Unit Ball
I. Chyzhykov and M. Voitovych
Faculty of Mechanics and Mathematics, Lviv Ivan Franko National University
1 Universytetska Str., Lviv 79000, Ukraine
E-mail: chyzhykov@yahoo.com
urkevych@gmail.com
Received September 22, 2014, revised January 14, 2015
The growth of analytic and harmonic functions in the unit ball Bn rep-
resented by the Cauchy–Stieltjes or Poisson–Stieltjes integral is studied.
A description of the growth is given in terms of smoothness of the Stieltjes
measure.
Key words: holomorphic function, Cauchy–Szegő transform, modulus
of continuity, Lipschitz class, Poisson integral, Cauchy integral, Cauchy–
Stieltjes integral, Poisson–Stielstjes integral, unit ball.
Mathematics Subject Classification 2010: 32A26 (primary); 32A25 (se-
condary).
1. Introduction
It is reasonable to expect for a function analytic in a bounded domain and
continuous up to the boundary to be smooth on the boundary if its derivative
grows slowly, and conversely. For the unit disk this was established by Hardy–
Littlewood ([8], [6, Chap. 5]).
We say that a complex-valued function f(eiθ), θ ∈ R is of the class Λ∗α (0 <
α ≤ 1) if ω∗(t) = O(tα) as t → 0 where ω∗(t) is the modulus of continuity of
f(eiθ), i.e.,
ω∗(t) = sup
|eiθ1−eiθ2 |≤t
|f(eiθ1)− f(eiθ2)|.
Theorem A ([6, Theorem 5.1]). Let f(z) be a function analytic in D =
{z ∈ C : |z| < 1}. Then f(z) is continuous in D and f(eiθ) ∈ Λ∗α (0 < α ≤ 1) if
and only if
f ′(z) = O
(
1
(1− |z|)1−α
)
.
c© I. Chyzhykov and M. Voitovych, 2015
On the Growth of the Cauchy–Szegő Transform in the Unit Ball
Let z, w ∈ Cn, n ∈ N, 〈z, w〉 =
∑n
i=1 ziwi, |z| = 〈z, z〉 1
2 . We denote by
Bn = {z ∈ Cn : |z| < 1} the unit ball in Cn and by Sn = {z ∈ Cn : |z| = 1} the
unit sphere.
For a complex-valued function f on Sn and a Borel measure µ on Sn, we
denote the Cauchy integral
C[f ](z) =
∫
Sn
f(ξ)dm2n−1(ξ)
(1− 〈z, ξ〉)n
, z ∈ Bn,
where m2n−1 is the normalized Lebesgue measure on Sn, m2n−1(Sn) = 1, and
C[µ](z) =
∫
Sn
dµ(ξ)
(1− 〈z, ξ〉)n
, z ∈ Bn (1)
the Cauchy–Stieltjes integral. Similarly, we denote by
P [f ](z) =
∫
Sn
(1− |z|2)n
|1− 〈z, ξ〉|2n
f(ξ)dm2n−1(ξ), z ∈ Bn,
P [µ](z) =
∫
Sn
(1− |z|2)n
|1− 〈z, ξ〉|2n
dµ(ξ), z ∈ Bn
the Poisson and Poisson–Stieltjes integrals, respectively.
Let f be a holomorphic function in Bn and f =
∑∞
k=0 Fk be the homogeneous
decomposition of f , then (Rf)(z) =
∑∞
k=0 kFk(z), z ∈ Bn is the radial derivative.
The following theorems were proved by W. Rudin for several complex variables.
Theorem B ([10]). Let 0 < α < 1 and f be a measurable complex func-
tion such that |f | is integrable with respect to the measure m2n−1 on Sn. Then
|f(eiθξ)− f(eitξ)| ≤ |eiθ − eit|α, ξ ∈ Sn, θ, t ∈ R, implies that
|(RC[f ])(z)| ≤ Aα(1− |z|)α−1, z ∈ Bn.
Theorem C ([10]). Let 0 < α < 1 and f be holomorphic in Bn. Then
|(RC[f ])(z)| ≤ (1− |z|)α−1, z ∈ Bn,
implies that f has a continuous extension to Bn which satisfies the Lipschitz
condition of order α.
Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 3 237
I. Chyzhykov and M. Voitovych
Some results in this direction that concern the unit polydisk can be found in [4,
5, 7]. In particular, necessary and sufficient conditions of the growth of Poisson–
Stieltjes integral in terms of Stieltjes measure were described in [3, 4]. Some
properties of harmonic functions of Lipschitz type spaces and their generalizations
are described in [1, 2]. In particular, a multidimensional counterpart of Theorem
A for harmonic functions in Bn was proved by S. Krantz in [9]. Note that, in
general, the functions represented by the Poisson–Stieltjes integral or the Cauchy–
Stieltjes integral can not be represented by the Poisson integral or the Cauchy
integral, respectively.
We are interested in the description of the growth of analytic and harmonic
functions in the unit ball Bn represented by the Cauchy–Stieltjes or the Poisson–
Stieltjes integral. The case of differentiable measures (with respect to m2n−1) is
well known (see, e.g., [10, Chap. 3] and [11, Chap. 7]). We find sharp estimates
for the growth of the Cauchy integral in the unit ball in Cn in terms of smoothness
of the Stieltjes measure.
Denote by
d(z, ζ) =
√
|1− 〈z, ζ〉|, z, ζ ∈ Bn,
the anisotropic metric on Sn ([10, Sec. 5.1]) and by
ω(δ, µ) = sup
z0∈Sn
|µ|({ξ ∈ Sn : d(ξ, z0) ≤ δ})
the modulus of continuity, where |µ| is the total variation of a complex-valued
Borel measure µ on Sn.
Theorem 1. Let µ be a complex-valued Borel measure on Sn, p ∈ (0, n].
Then
∃c > 0 ω(δ, µ) ≤ cδ2(n−p), 0 < δ ≤
√
2,
implies that
C[µ](z) = O
(
1
(1− |z|)p
)
, z ∈ Bn.
The examples in Sec. 3 show that the estimate is sharp up to a constant
factor. In order to prove Theorem 1, we use the standard approach [10]. The
same method allows us to prove a criterion for the Poisson integral.
Theorem 2. Let µ be a positive Borel measure on Sn, p ∈ (0, n). Then
∃c > 0 ω(δ, µ) ≤ cδ2(n−p), 0 < δ < 1 ⇔ P [µ](z) = O
(
1
(1− |z|)p
)
, z ∈ Bn.
238 Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 3
On the Growth of the Cauchy–Szegő Transform in the Unit Ball
R e m a r k. Theorems 1 and 2 could be easily generalized for the integrals
with kernels of the form
1
(1− 〈z, ξ〉)n+s
,
1
|1− 〈z, ξ〉|n+s
, s ∈ R,
respectively, for an appropriate choice of p.
2. Proofs of the Theorems
P r o o f of Theorem 1. Denote
Ek(z) =
{
ξ ∈ Sn :
∣∣∣∣1− 〈
z
|z| , ξ〉
∣∣∣∣ < 2k+1(1− |z|)
}
for z ∈ Bn \ {0} and k ∈ {0, 1, 2, . . .}. Then (E−1(z) := ∅)
∞⋃
k=0
(Ek(z) \ Ek−1(z)) = Sn.
Since ∀k ∈ N ∀ξ ∈ Ek(z) \ Ek−1(z) ∀z : 1 > |z| > 3
4
|1− 〈z, ξ〉| ≥ ||1− |z|| − ||z| − 〈z, ξ〉|| = |z|
∣∣∣∣1−
〈 z
|z| , ξ
〉∣∣∣∣− (1− |z|)
≥ |z|2k(1− |z|)− (1− |z|) = (|z|2k − 1)(1− |z|)
and ∀ξ ∈ E0(z): |1− 〈z, ξ〉| ≥ 1− |〈z, ξ〉| ≥ 1− |z|, we have
|C[µ](z)| =
∣∣∣∣∣∣
∫
Sn
dµ(ξ)
(1− 〈z, ξ〉)n
∣∣∣∣∣∣
=
∣∣∣∣∣∣∣
∞∑
k=1
∫
Ek(z)\Ek−1(z)
dµ(ξ)
(1− 〈z, ξ〉)n
+
∫
E0(z)
dµ(ξ)
(1− 〈z, ξ〉)n
∣∣∣∣∣∣∣
≤
∞∑
k=1
∫
Ek(z)\Ek−1(z)
|dµ(ξ)|
(|z|2k − 1)n(1− |z|)n
+
∫
E0(z)
|dµ(ξ)|
(1− |z|)n
≤ (1− |z|)−n
∞∑
k=1
(|z|2k − 1)−n|µ|(Ek(z)) + (1− |z|)−n|µ|(E0(z))
≤ (1− |z|)−n
∞∑
k=1
ω(
√
2k+1(1− |z|), µ)
(|z|2k − 1)n
+ (1− |z|)−nω(
√
2(1− |z|) , µ)
Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 3 239
I. Chyzhykov and M. Voitovych
≤ (1− |z|)−n
∞∑
k=1
(
3
4
2k − 1
)−n
c(2k+1(1− |z|))n−p + 2n−pc(1− |z|)−p
<
c
(1− |z|)p
(
2n−p
∞∑
k=1
2k(n−p)
(3 · 2k−2 − 1)n
+ 2n−p
)
,
3
4
< |z| < 1.
Since the last series is convergent, we get the desired result.
Now, let |z| ≤ 3
4 . Since d(1, z) ≤ √
2 for all z ∈ Bn,
|µ|(Sn) ≤ ω(
√
2, µ) ≤ c2n−p.
Then
|C[µ](z)| ≤
∫
Sn
∣∣∣∣
dµ(ξ)
(1− 〈z, ξ〉)n
∣∣∣∣ ≤
|µ|(Sn)
(1− |z|)n
≤ c2n−p
(1− |z|)p
(
1
4
)n−p ≤
c8n−p
(1− |z|)p
.
P r o o f of Theorem 2.(⇐) For all ξ ∈ E1(z),
|1− 〈z, ξ〉| ≤
∣∣∣∣1−
〈 z
|z| , ξ
〉∣∣∣∣ +
∣∣∣∣
〈 z
|z| − z, ξ
〉∣∣∣∣ ≤ 4(1− |z|) +
∣∣∣∣
z
|z| − z
∣∣∣∣ = 5(1− |z|).
By the assumption ∃c > 0 such that
c
(1− |z|)p
≥
∣∣∣∣∣∣
∫
Sn
(1− |z|2)n
|1− 〈z, ξ〉|2n
dµ(ξ)
∣∣∣∣∣∣
≥
∣∣∣∣∣∣∣
∫
E1(z)
(1− |z|2)n
|1− 〈z, ξ〉|2n
dµ(ξ)
∣∣∣∣∣∣∣
≥ (1 + |z|)n
52n(1− |z|)n
µ(E1(z)). (2)
Since for d(z, ξ) <
√
3(1− |z|) implies
∣∣∣∣1−
〈 z
|z| , ξ
〉∣∣∣∣ ≤ |1− 〈z, ξ〉|+
∣∣∣∣〈z, ξ〉 − 〈 z
|z| , ξ
〉∣∣∣∣
< 3(1− |z|) +
∣∣∣∣z −
z
|z|
∣∣∣∣ = 4(1− |z|),
we get E1(z) ⊃ {ξ ∈ Sn : d(z, ξ) <
√
3(1− |z|)}.
240 Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 3
On the Growth of the Cauchy–Szegő Transform in the Unit Ball
From inequality (2) and the last inclusion it follows that
µ(E1(z)) ≤ c1(1− |z|)n−p, z ∈ Bn,
ω(
√
3(1− |z|), µ) ≤ c1(1− |z|)n−p, z ∈ Bn,
ω(δ, µ) ≤ c1δ
2(n−p)3p−n, 0 < δ ≤
√
3,
where c1 ≥ 52nc/2n.
(⇒) Using the arguments similar to those of the proof of Theorem 1, we get
|P [µ](z)| =
∫
Sn
(1− |z|2)ndµ(ξ)
|1− 〈z, ξ〉|2n
=
∞∑
k=1
∫
Ek(z)\Ek−1(z)
(1− |z|2)ndµ(ξ)
|1− 〈z, ξ〉|2n
+
∫
E0(z)
(1− |z|2)ndµ(ξ)
|1− 〈z, ξ〉|2n
≤
∞∑
k=1
∫
Ek(z)\Ek−1(z)
(1− |z|2)ndµ(ξ)
(|z|2k − 1)2n(1− |z|)2n
+
∫
E0(z)
(1− |z|2)ndµ(ξ)
(1− |z|)n
≤ (1− |z|)−n
∞∑
k=1
(1 + |z|)nµ(Ek(z))
(|z|2k − 1)2n
+ (1− |z|)−n(1 + |z|)nµ(E0(z))
≤ (1− |z|)−n
∞∑
k=1
2nω(
√
2k+1(1− |z|), µ)
(|z|2k − 1)2n
+ (1− |z|)−n2nω(
√
2(1− |z|), µ)
≤ (1− |z|)−n
∞∑
k=1
(
3
4
2k − 1
)−2n
2nc(2k+1(1− |z|))n−p + 2n−p2nc(1− |z|)−p
≤ c
(1− |z|)p
(
22n−p
∞∑
k=1
2k(n−p)
(3 · 2k−2 − 1)2n
+ 22n−p
)
.
The convergence of the last series implies the required inequality. If |z| ≤ 3
4 ,
using the arguments similar to those of the proof of Theorem 1, we can obtain
|P [µ](z)| ≤
∫
Sn
∣∣∣∣
(1− |z|2)ndµ(ξ)
|1− 〈z, ξ〉|2n
∣∣∣∣ ≤
2n|µ|(Sn)
(1− |z|)n
≤ c2n−p2n
(1− |z|)p
(
1
4
)n−p ≤
c24n−3p
(1− |z|)p
.
3. Examples
1. Let in Theorem 1 µ be the Lebesgue measure m2n−1 on Sn. Note that
Qδ = {ξ ∈ Sn : d(ξ, z0) < δ} is a “ball” on Sn and m2n−1(Qδ) ³ δ2n, δ → 0
Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 3 241
I. Chyzhykov and M. Voitovych
([10, Ch. 5]). Since “the center” z0 is of no particular importance, the modulus
of continuity
ω(δ,m2n−1) = sup
z0∈Sn
m2n−1({ξ ∈ Sn : d(ξ, z0) < δ}) ³ δ2n, δ → 0,
so ([10, Prop. 1.4.10])
∫
Sn
dm2n−1(ξ)
|1− 〈z, ξ〉|n ³ ln
1
1− |z| , |z| ↑ 1.
Hence the statement of Theorem 1 is not true for the case p = 0.
2. Let µ = δξ0 , i.e.,
µ(A) =
{
c, A 3 ξ0;
0, otherwise,
where A ⊂ Sn, ξ0 ∈ Sn. Then
ω(δ, µ) = c ∈ C, p = n
and
C[µ](tξ0) =
∫
Sn
dµ(ξ)
(1− 〈tξ0, ξ〉)n
= c
1
(1− t)n
, 0 < t < 1.
This example shows the sharpness of Theorem 1 for p = n.
3. Let µ be a Borel measure on S2 ⊂ C2 and
µ(ξ) =
{
k−l, ξ = (1− k−q,
√
1− (1− k−q)2)
0, otherwise,
where 1 < l < 2q + 1, k = 0, 1, . . . .
Then ω(δ, µ) ³ δ
2 l−1
q , i.e., µ ∈ Λ l−1
q
and max
|z|=r
|C[µ](z)| ≥ c
(
1
(1−r)
2− l−1
q
)
.
Indeed,
ω(δ, µ) =
∫
|1−ξ1|<δ2
dµ(ξ1, ξ2) =
∑
k−q<δ2
k−l =
∑
k>δ
− 2
q
k−l ³ δ
2
q
(l−1)
.
Let z = re1, where e1 = (1, 0) ∈ S2, r ∈ (0, 1),
C[µ](re1) =
∫
S2
dµ(ξ)
(1− rξ1)2
=
∞∑
k=1
k−l
(1− r(1− k−q))2
242 Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 3
On the Growth of the Cauchy–Szegő Transform in the Unit Ball
≥
2
[
1
(1−r)1/q
]
∑
k=
[
1
(1−r)1/q
]
1
kl(1− r + r
kq )2
≥
2
[
1
(1−r)1/q
]
∑
k=
[
1
(1−r)1/q
]
(1− r)
l
q
2l
1
((1− r) + r(1− r))2
≥ 1
2(1− r)
1
q
(1− r)
l
q
2l
1
4(1− r)2
=
1
2l+3
(1− r)
l−1
q
−2
.
4. The Operator Theory Point of View
Equality (1) is often considered as the Cauchy–Szegő operator acting from
the space M(Sn) (M+(Sn)) of Borel (positive) measures on Sn into the class of
analytic functions on Bn.
Denote by Hp
q (Bn), 1 ≤ p ≤ ∞, q ≥ 0, the class of analytic functions f on
Bn with the norm
‖f‖p
q = sup
0<r<1
(1− r)q
∫
Sn
‖f(rξ)‖pdm2n−1(ξ)
1/p
,
‖f‖∞q = sup
0<r<1
(1− r)q max
|ξ|=r
|f(ξ)|.
Also denote by hp
q the class of harmonic functions with the same norm. It is
known that there exist the measures µ ∈ M(Sn) such that C[µ] /∈ H1
0 (Bn). If
we denote by Λα(Sn) (Λ+
α (Sn)) the class of (positive) measures on Sn such that
‖µ‖α = sup
0<δ≤√2
ω(δ,µ)
δ2α < +∞, it then follows from Theorems 1 and 2 that C[µ]
and P [µ] are bounded operators from Λα(Sn) ⊂ M(Sn) and Λ+
α (Sn) ⊂ M+(Sn)
into H∞
n−α(Bn) and h∞n−α(Bn), respectively. Moreover,
‖C‖ = sup
|µ|6=0
‖Cµ‖∞n−α
‖µ‖α
≤ max
{
2n−p
∞∑
k=1
2k(n−p)
(3 · 2k−2 − 1)n
+ 2n−p, 8n−p
}
≤ max
{
2n−p
∞∑
k=1
2k(n−p)
2(k−2)n
+ 2n−p, 8n−p
}
≤ max
{
23n
2p − 1
+ 2n−p, 8n−p
}
=
23n
2p − 1
+ 2n−p
Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 3 243
I. Chyzhykov and M. Voitovych
and
‖P‖ = sup
|µ|6=0
‖Pµ‖∞n−α
‖µ‖α
≤ max
{
22n−p
∞∑
k=1
2k(n−p)
(3 · 2k−2 − 1)2n
+ 22n−p, 24n−3p
}
≤ max
{
27n
2n+p − 1
+ 22n−p, 24n−3p
}
=
27n
2n+p − 1
+ 22n−p.
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