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Journal ArticleDOI

On subadditivity of analytic capacity for two continua

01 Jan 1984-Kodai Mathematical Journal (Department of Mathematics, Tokyo Institute of Technology)-Vol. 7, Iss: 1, pp 73-75

About: This article is published in Kodai Mathematical Journal.The article was published on 1984-01-01 and is currently open access. It has received 10 citation(s) till now. The article focuses on the topic(s): Thesaurus (information retrieval) & Subadditivity.

Summary (1 min read)

Jump to: [ll/llsi] and [2. Extremal problem.]

ll/llsi

  • The following problem was first raised by Vituskin [6] .
  • This shows that the constant M possibly depends on positions of two sets.

2. Extremal problem.

  • For simplicity's sake the authors write K lf K 2 for its boundary components which have the same analytic capacities as the original continua.
  • Since the analytic capacity is invariant under a conformal mapping with the normalization (4), the authors have Since the analytic capacity is equal to the capacity for a continuum [7] , they obtain the assertion from the minimum property in Theorem 1.

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N.
SUITA
KODAI
MATH.
J.
7 (1894), 73-75
ON
SUBADDITIVITY
OF
ANALYTIC
CAPACITY
FOR TWO CONTINUA
Dedicated
to
Professor Mitsuru Ozawa
on his
60th birthday
BY
NOBUYUKI
SUITA
1.
Introduction.
Let
K be a
compact
set in
the complex plane
C
and
let
Ω(K)
be
the connected component
of
C—K containing the point
at
infinity.
The
analytic capacity γ(K)
of K is
defined
by
r(/O=sup
|/'(oo)|.
ll/llsi
Here
f(z) is a
holomorphic function
in
Ω(K) whose expansion
at the
point
at
infinity
is
given
by
f(z)=f'{°o)/z+a
s
/z*+~'
and
11/11 denotes the supnorm
of /.
The
following problem
was
first
raised
by
Vituskin [6]. For two disjoint
compact
sets
K
λ
and
K
2
, is
there an absolute constant
M
such that the following
semiadditivity holds
We
can
find this problem
and
partial answers
in
many articles
and
books
[1,
2, 3, 6,
7].
It
was shown
by
MeΓnikov
[3]
that
if K
1
and
K
2
are
separated
by
an
analytic curve
c,
then there
exists
a
constant M(c) depending
on c for
which (1) holds. This shows that the constant
M
possibly depends
on
positions
of two sets.
In
the present paper
we
shall establish the following subadditive inequality
(2)
riK^Kj^γiKj+ΠKz)
for arbitrary disjoint continua
K
λ
and
K
2
. To
prove (2)
we
shall
solve
an
auxi-
liary extremal problem.
2.
Extremal
problem.
Let Ω be a
doubly connected domain containing the
point
at
infinity.
We
suppose that its boundary components are continua
K
Y
and
Received March 22, 1983
73

74 NOBUYUKI SUITA
K
2
. Let EF be the family of univalent functions φ defined on Ω with a normal-
ization
(3) φ(z)=z+ajz+••• at co.
We denote by φ(Kj) the boundary component of the image domain φ{Ω) under φ
which corresponds to Kj (j=l, 2). Our problem is to minimize Cap ^(ϋΓJ+Cap φ(K
2
)
within £F, Cap φ(Kj) being logarithmic capacity of φ{K
3
). We prove
THEOREMI.
The minimum of the quantity Cap 0(AΊ)+Cap φ(K
2
) is
attained
by the con
formal
mapping
φ
0
which
maps
Ω
onto
a
parallel
slit
domain Δ
whose
boundary
lies
on a
straight
line.
Proof.
Since capacity is invariant under rigid motions, we consider the same
problem within a slightly larger class § of univalent functions φ whose expansion
admits
(4)
φ{z)
= e
z+a<>+ajz+- at co.
There
exists
a unique radial
slit
disc mapping φ(z) in 5 such that φ{K
±
) is a
disc
I
w
I
Sr, say and ψ(K
2
) is a line segment a^w^b (a>r). Then Cap φ{K^)=r
and
we know that r is equal to the minimum of Cap
φ{K^).
For these facts see
Sario and Oikawa ([5], pp. 148-158). By an elementary mapping S
r
{w)
r(w/r+r/w), ψ(K
λ
) is mapped onto a line segment [—2r, 2r] with the same
capacity r. Then ψ{K
2
) is mapped onto a new segment [α',
b'~\.
We show that
Cap[α',
6']=:(&'—αO/4 is the minimum of Cap 0(iQ within £F. In fact by a
parallel displacement, [α', fr'] is moved onto
[—2r',
2r'], r
f
—{b
f
—a
f
)/A. The
conformal mapping S^
1
maps the last domain onto another radial
slit
disc, K
2
corresponding to the disc
\w\^r
f
.
Hence r' is the minimum of Cap
φ(K
2
).
Thus
r+r
/
is the desired minimum within §.
S
r
°ψ
is clearly an extremal function.
It
is easy to see that by composition of elementary mappings an extremal func-
tion
φ
0
can be constructed in £F as stated in the theorem, which completes the
proof.
Remark. The uniqueness of the extremal function in 9" is deduced from
that
of a radial
slit
disc mapping. Indeed the radial
slit
disc mapping ψ with
lϊm
z
^φ(z)/z—l is a unique extremal mapping minimizing radii within the family
of univalent functions φ
satisfying
Iim
2
^oo^(^)/z=l and that φ{K
λ
) is a disc
centered
at the origin. It is not difficult to derive the uniqueness directly by
means
of the method of extremal metrics.
3.
Subadditivity.
We prove
THEOREM
2. // K
x
and K
2
are mutually
disjoint
continua, then the
inequality
(2)
holds.

ON
SUBADDITIVITY
OF
ANALYTIC CAPACITY
75
Proof.
If Ω(Kj)aΩ(K
k
) (jΦk), the inequality is trivial. Otherwise Ω(K^JK
2
)
forms a doubly connected domain. For simplicity's
sake
we write K
lf
K
2
for its
boundary components which have the same analytic capacities as the original
continua.
Let φ
0
be the extremal mapping for this domain in Theorem 1.
Pommerenke
showed that for a compact set on a straight line its analytic capacity
is equal to a quater of its total length [4]. Since the analytic capacity is
invariant under a conformal mapping with the normalization (4), we have
Since the analytic capacity is equal to the capacity for a continuum [7], we
obtain
the assertion from the minimum property in Theorem 1.
REFERENCES
[ 1 ]
BRAXNNAN,
D.
A.
AND
J. C.
CLUNIE,
Aspect
of
contemporary complex analysis,
Academic Press (1980).
[2]
GARNETT,
J.,
Analytic capacity
and
measure, Springer Lecture Notes
in
Math.
297 Springer-Verlag (1972).
[ 3 ]
MEL'NIKOV,
M.
S.,
Estimate
of the
Cauchy integral along
an
analytic curve, Math.
Sbornik
71
(1966), 503-515 (Russian), Amer. Math.
Soc.
Translation
ser. 2, 80
(1969), 243-256.
[4]
POMMERENKE,
C,
Uber
die
analytische Kapatitat, Arch,
der
Math.
11
(1960),
270-277.
[ 5 ]
SARIO,
L.
AND
K.
OIKAWA,
Capacity functions, Springer-Verlag (1969).
[6]
VITUSKIN,
A.
G.,
The
analytic capacity
of
sets
in
problems
of
approximation
theory, Uspehi
Mat.
Nauk
22
(1967), 141-199 (Russian), Russian Math,
surveys
22 (1967), 139-200.
[7]
ZALCMAN,
L.,
Analytic capacity
and
rational approximation, Springer Lecture
Notes
in
Math.
50,
Springer-Verlag (1968).
DEPARTMENT
OF
MATHEMATICS
TOKYO
INSTITUTE
OF
TECHNOLOGY
Citations
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Journal ArticleDOI
Xavier Tolsa1Institutions (1)
01 Mar 2003-Acta Mathematica
Abstract: Let $\gamma(E)$ be the analytic capacity of a compact set $E$ and let $\gamma_+(E)$ be the capacity of $E$ originated by Cauchy transforms of positive measures. In this paper we prove that $\gamma(E)\approx\gamma_+(E)$ with estimates independent of $E$. As a corollary, we characterize removable singularities for bounded analytic functions in terms of curvature of measures, and we deduce that $\gamma$ is semiadditive, which solves a long standing question of Vitushkin.

270 citations


Posted Content
Xavier Tolsa1Institutions (1)
Abstract: Let $\gamma(E)$ be the analytic capacity of a compact set $E$ and let $\gamma_+(E)$ be the capacity of $E$ originated by Cauchy transforms of positive measures. In this paper we prove that $\gamma(E)\approx\gamma_+(E)$ with estimates independent of $E$. As a corollary, we characterize removable singularities for bounded analytic functions in terms of curvature of measures, and we deduce that $\gamma$ is semiadditive, which solves a long standing question of Vitushkin.

227 citations


Cites background from "On subadditivity of analytic capaci..."

  • ...This question was raised by Vitushkin in the early 1960's (see [Vii and [VIM]) and was known to be true only in some particular cases (see [Me1] and [ Su ] for example, and [De] and [DeO] for some related results)....

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Journal ArticleDOI
Abstract: 1. Introduction. Let µ be a continuous (i.e., without atoms) positive Radon measure on the complex plane. The truncated Cauchy integral of a compactly supported function f in L p (µ), 1 ≤ p ≤ +∞, is defined by Ꮿ ε f (z) = |ξ −z|>ε f (ξ) ξ − z dµ(ξ), z ∈ C, ε > 0. In this paper, we consider the problem of describing in geometric terms those measures µ for which |Ꮿ ε f | 2 dµ ≤ C |f | 2 dµ, (1) for all (compactly supported) functions f ∈ L 2 (µ) and some constant C independent of ε > 0. If (1) holds, then we say, following David and Semmes [DS2, pp. 7–8], that the Cauchy integral is bounded on L 2 (µ). A special instance to which classical methods apply occurs when µ satisfies the doubling condition µ(2) ≤ Cµ((), for all discs centered at some point of spt(µ), where 2 is the disc concentric with of double radius. In this case, standard Calderón-Zygmund theory shows that (1) is equivalent to Ꮿ * f 2 dµ ≤ C |f | 2 dµ, (2) where Ꮿ * f (z) = sup ε>0 |Ꮿ ε f (z)|. If, moreover, one can find a dense subset of L 2 (µ) for which Ꮿf (z) = lim ε→0 Ꮿ ε f (z) (3) 269 270 XAVIER TOLSA exists a.e. (µ) (i.e., almost everywhere with respect to µ), then (2) implies the a.e. (µ) existence of (3), for any f ∈ L 2 (µ), and |Ꮿf | 2 dµ ≤ C |f | 2 dµ, for any function f ∈ L 2 (µ) and some constant C. For a general µ, we do not know if the limit in (3) exists for f ∈ L 2 (µ) and almost all (µ) z ∈ C. This is why we emphasize the role of the truncated operators Ꮿ ε. Proving (1) for particular choices of µ has been a relevant theme in classical analysis in the last thirty years. Calderón's paper [Ca] is devoted to the proof of (1) when µ is the arc length on a Lipschitz graph with small Lipschitz constant. The result for a general Lipschitz graph was obtained by Coifman, McIntosh, and Meyer in 1982 in the celebrated paper [CMM]. The rectifiable curves , for which (1) holds for the arc length measure µ on …

120 citations


01 Jan 2006-
Abstract: A compact set E �¼ C is said to be removable for bounded analytic functions if for any open set  containing E, every bounded function analytic on  \ E has an analytic extension to . Analytic capacity is a notion that, in a sense, measures the size of a set as a non removable singularity. In particular, a compact set is removable if and only if its analytic capacity vanishes. The so-called Painleve problem consists in characterizing removable sets in geometric terms. Recently many results in connection with this very old and challenging problem have been obtained. Moreover, it has also been proved that analytic capacity is semiadditive. We review these results and other related questions dealing with rectifiability, the Cauchy transform, and the Riesz transforms.

30 citations


Journal ArticleDOI
Malik Younsi1, Thomas Ransford1Institutions (1)
Abstract: We develop a least-squares method for computing the analytic capacity of compact plane sets with piecewise-analytic boundary. The method furnishes rigorous upper and lower bounds which converge to the true value of the capacity. Several illustrative examples are presented. We are led to formulate a conjecture which, if true, would imply that analytic capacity is subadditive. The conjecture is proved in a special case.

8 citations


Cites background from "On subadditivity of analytic capaci..."

  • ...The subadditivity problem for analytic capacity This section is about the study of the following question: Is it true that (6) γ(E ∪ F ) ≤ γ(E) + γ(F ) for all compact sets E,F? Suita [19] proved that (6) holds if E,F are disjoint connected compact sets....

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References
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Book
01 Jan 1972-
Abstract: Analytic capacity.- The cauchy transform.- Hausdorff measure.- Some examples.- Applications to approximation.

257 citations


Book
01 Jan 1968-
Abstract: Peak points.- Analytic capacity.- Some useful facts.- Estimates for integrals.- Melnikov's theorem.- Further results.- Applications.- The problem of rational approximation.- AC capacity.- A scheme for approximation.- Vitushkin's theorem.- Applications of Vitushkin's theorem.- Geometric conditions.- Function algebra methods.- Some open questions.

108 citations


Journal ArticleDOI

31 citations


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