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MATHEMATICS
A. G. VITUSHKIN
ANALYTIC CAPACITY OF SETS
AND SOME OF ITS PROPERTIES
(Presented by Academician A. N. Kolmogorov, 8 X 1958)
The concept of the analytic capacity of a set arose in connection with the study of sets on which every continuous function can be expanded in a uniformly convergent series of rational functions. The results pertaining to this problem will be published later; here only some preliminary propositions will be formulated.
§ 1. Definition of analytic capacity. Let \(e\) be some bounded set lying in the complex plane \(\tau\); let \(\varphi(z)\) be a complex function of the complex variable \(z=x+iy\), possessing the following properties: a) the function \(\varphi(z)\) is defined everywhere on \(\tau-e\), and \(|\varphi(z)| \leqslant 1\); b) \(\varphi(z)\) is analytic everywhere outside the set \(e\), i.e., for every point of \(\tau-e\) there exists a neighborhood in which \(\varphi(z)\) is expandable in a uniformly convergent, throughout the whole neighborhood, series of polynomials; c) \(\lim\limits_{z\to\infty}\varphi(z)=0\).
Put \(\gamma(e,\varphi)=\lim\limits_{z\to\infty}|z\varphi(z)|\) and \(\gamma(e)=\sup\limits_{\varphi}\gamma(e,\varphi)\) (the least upper bound is taken over all functions \(\varphi(z)\) satisfying conditions a), b), c)).
The felicitous name for the quantity \(\gamma(e)\) was devised by V. D. Erokhin—he proposed calling the quantity \(\gamma(e)\) the analytic capacity of the set \(e\).
As an explanation of item b), we note that for every \(\varepsilon>0\) one can specify a closed set \(e_\varepsilon \subset e\) and a function \(\varphi_e^\varepsilon(z)\), satisfying conditions a), b), c) (with respect to \(e_\varepsilon\)), such that \(\gamma(e_\varepsilon)\geqslant \gamma(e)-\varepsilon\) and \(\lim\limits_{z\to\infty} z\varphi_e^\varepsilon(z)=\gamma_\varepsilon\geqslant \gamma(e)-\varepsilon\), where \(\gamma_\varepsilon\) is a real number. Put \(\varphi_e(z)=\lim\limits_{\varepsilon\to0}\varphi_e^\varepsilon(z)\). We shall call this function \(\varphi_e(z)\), for the set \(e\), the Ahlfors function.
Some of the simplest properties of \(\gamma(e)\) and \(\varphi_e\):
-
For every set \(e\), \(\gamma(e)\geqslant 0\).
-
If \(e''\supset e'\), then \(\gamma(e'')\geqslant \gamma(e')\).
-
If the sets \(e'\) and \(e''\) are congruent, then \(\gamma(e')=\gamma(e'')\).
-
For every closed set \(e\), \(\gamma(e)=\gamma(\widetilde e)\), where \(\widetilde e\) is the boundary of the set \(e\).
-
The function \(\gamma(e)\) has dimension 1, i.e., under a dilation of the set by a factor of \(k\), its analytic capacity increases by a factor of \(k\).
-
For every set \(e\),
\[ \gamma(e)=\frac{1}{2\pi}h_1(e), \]
where \(h_1(e)\) is the Hausdorff length of the set \(e\). -
For every set \(e\) and \(\varepsilon>0\), one can specify a closed (in the plane) set \(e'\subset e\) such that \(\gamma(e')\geqslant \gamma(e)-\varepsilon\).
-
If the set \(e\) is closed, then \(\varphi_e(z)\) satisfies conditions a), b), c), and every function \(\varphi(z)\) satisfying conditions a), b), c) (with respect to \(e\)), for which \(\lim\limits_{z\to\infty} z\varphi(z)=\gamma(e)\), is identically equal (outside the set \(e\)) to the function \(\varphi_e(z)\).
-
If the set \(e\) consists of \(n\) closed components distinct from a point, then, by Ahlfors’ theorem, the function \(\varphi_e(z)\) assumes the value zero exactly \(n\) times (counting multiplicities of zeros).
-
If the set \(e\) consists of \(n\) closed components distinct from a point, then
\[ \gamma(e)=C(e)\prod_{k=1}^{n-1} e^{-h_k}\leqslant C(e), \]
where \(C(e)\) is the harmonic capacity of the set \(e\), and \(h_k>0\) are the values of the Green function \(g_e(\infty,z)\) at the zeros \(\{z_k\}\), distinct from \(z=\infty\), of the Ahlfors function \(\varphi_e(z)\). -
If \(e\) is a continuum, then \(\gamma(e)=C(e)\), since \(-\lg|\varphi_e(z)|\) coincides with the Green function \(g_e(\infty,z)\) of the set \(e\).
-
If \(e\) is a continuum, then \(A\gamma(e)\leqslant d(e)\leqslant B\gamma(e)\), where \(d(e)\) is the diameter of the set \(e\), and \(A>0\) and \(B>0\) are absolute constants.
-
If the planar measure of the set \(e\) is equal to \(s\), then \(\gamma(e)\geqslant \sqrt{s}/2\sqrt{\pi}\).
§ 2. Estimation, in terms of \(\gamma(e)\), of the coefficients of the expansion of an analytic function.
Lemma 1. Let \(g\) be some domain; let \(e\) be a subset of this domain at a positive distance from its boundary. Then every function \(f(z)\), analytic in \(g-e\) and bounded by the constant \(m\), can be represented in the form \(f(z)=C+\varphi(z)+g(z)\), where \(C\) is a constant; \(\varphi(z)\) is a function analytic outside the set \(e\), uniformly bounded (in modulus) by the constant \(\beta m\) and equal to zero at \(z=\infty\); \(g(z)\) is a function analytic in the domain \(g\), also bounded by the constant \(\beta m\) (\(\beta>0\) is a constant depending only on \(g\) and \(e\)).
Let us note that if \(g-e\) contains an annulus bounded by two concentric circles with ratio of radii \(\rho<1\), then
\[
\beta\leqslant \frac{2}{1-\rho}.
\]
Lemma 2. Every set \(e\) can be covered by a closed disk \(g_e\) with radius \(r_e\leqslant r_0 d(e)\), where \(r_0<1\) is an absolute constant, and \(d(e)\) is the one-dimensional diameter of the set \(e\).
Theorem 1. If the function \(\varphi(z)\) satisfies conditions a), b), c), then outside the disk of minimal radius containing the set \(e\), it is representable in the form of the series
\[
\varphi(z)=\sum_{k=1}^{\infty}\frac{c_k}{(z-a)^k},
\]
where \(a\) is the center of the mentioned disk, and the coefficients \(c_k\) satisfy the inequality
\[
|c_k|\leqslant A\gamma(e)[d(e)]^{k-1}
\]
(\(A>0\) is an absolute constant).
Proof. Consider the annulus bounded by the concentric circles \(K_1\{|z-a|=r_e\}\) and \(K_2\{|z-a|=\tfrac12[d(e)+r_e]\}\). For \(|z-a|\geqslant d(e)\) we have
\[
\begin{aligned}
|f(z)|&=\frac{1}{2\pi}\left|\int_{K_2}\frac{f(\xi)}{\xi-z}\,d\xi\right|
=\frac{1}{2\pi}\left|\int_{K_2}[c_z+\varphi_z(\xi)+g_z(\xi)]\,d\xi\right|\\
&=\frac{1}{2\pi}\left|\int_{K_2}\varphi_z(\xi)\,d\xi\right|
\leqslant \frac{1}{2\pi}\max_{K_2}|\varphi_z(\xi)|
\left|\int_{K_2}\frac{\varphi_z(\xi)\,d\xi}{\max|\varphi_z(\xi)|}\right|\\
&\leqslant \max|\varphi_z(\xi)|\,\gamma(e)
\leqslant \beta m_z\gamma(e)
\leqslant A\frac{\gamma(e)}{d(e)}.
\end{aligned}
\]
Consequently,
\[
|c_k|=\frac{1}{2\pi}\left|\int_{|z-a|=d(e)} f(z)(z-a)^{k-1}\,dz\right|
\leqslant A\gamma(e)[d(e)]^{k-1}.
\]
Corollary 1. In order that every bounded function analytic outside the set \(e\) be constant, it is necessary and sufficient that the analytic capacity of the set \(e\) be equal to zero.
Corollary 2. Let \(g\) be some domain with rectifiable boundary \(\gamma\), and let \(e \subset g\) be a set at a positive distance from \(\gamma\). Then for every function \(f(z)\) analytic in \(g-e\) the inequality
\[ \left|\int_\gamma f(z)\,dz\right| \leq B \max_{g-e}|f(z)|\,\gamma(e), \]
holds, where \(B>0\) is an absolute constant.
§ 3. Condensation points of a set. Some notation: \(Ce\) is the complement of the set \(e\) in the whole plane \(\tau\); \(\sigma_z^r\) is the open disk of radius \(r\) with center at the point \(z\); \(K_z^n\) is the closed annulus bounded by two circles with radii \(r_n=1/2^n\) and \(r_{n+1}=1/2^{n+1}\) with common center at the point \(z\); \(P_z^r(e)=\dfrac{1}{r}\gamma(e\cap\sigma_z^r)\) is the mean analytic density of the set \(e\) in the disk \(\sigma_z^r\); \(P_z(e)=\inf_{r\to0} P_z^r(e)\) is the analytic density of the set \(e\) at the point \(z\); \(e_\infty\) is the set of all such points \(z\) of the plane \(\tau\) for each of which
\[ \sup_{r\to0}\frac{1}{r^2}\gamma(Ce\cap\sigma_z^r)=\infty. \]
Definition. A point \(z\in\tau\) will be called a condensation point of the set \(e\) if
\[ \sum_{n=1}^{\infty}\left(\frac{1}{r_n}\right)^2\gamma(Ce\cap K_z^n)<\infty. \]
If the indicated series diverges, then the point \(z\) will be called a rarefaction point of the set \(e\).
Lemma 3. If the set \(e\) is a simply connected domain and \(d(e)<1\), then the quantity
\[ r_z(e)=\left[\sum_{n=1}^{\infty}4^n\gamma(Ce\cap K_z^n)\right]^{-1} \]
is comparable with the distance \(\rho(z,Ce)\) from the point \(z\) to the set \(Ce\), i.e. there exist positive absolute constants \(C'\), \(C''\) such that
\[ C'\rho(z,Ce)\leq r_z(e)\leq C''\rho(z,Ce). \]
From this lemma, in particular, it follows that every interior point of the set \(e\) is its condensation point.
Lemma 4. For every set \(e\) and disk \(\sigma_z^r\) the inequality
\[ P_z^r(e)\geq C_1\frac{s}{r^2}, \]
holds, where \(s\) is the exterior planar measure of the set \(e_\infty\cap\sigma_z^r\), and \(C_1>0\) is an absolute constant.
Proof. Cover the set \(e_\infty\cap\sigma_z^r\) by a system of disks \(\{\sigma_{z_k}^{r_k}\}\) such that
\[ \gamma(Ce\cap\sigma_{z_k}^{r_k})\geq \frac{1}{r}(r_k)^2. \]
From this system choose a finite subsystem of disks \(\sigma_k=\sigma_{z_k}^{r_k}\) \((k=1,2,\ldots,n)\) such that
\[ s \geq \sum_{k=1}^{n}\pi(r_k)^2 \geq C_2s \]
(\(C_2>0\) is some sufficiently small, but absolute constant) and such that the disks \(\sigma'_k=\sigma_{z_k}^{4r_k}\) \((k=1,2,\ldots,n)\) are pairwise disjoint. For each of the sets \(\{Ce\cap\sigma_k\}\) fix a function \(\varphi_k(\xi)\), analytic outside the set \(Ce\cap\sigma_k\), bounded (in modulus) by the constant
\[ \frac{(r_k)^2}{r\,\gamma(Ce\cap\sigma_k)} \]
and such that
\[ \lim_{\xi\to\infty}\xi\varphi_k(\xi)=\frac{1}{2r}(r_k)^2. \]
Consider the function
\[ \varphi(\xi)=\frac{1}{1+m}\sum_{k=1}^{n}\varphi_k(\xi), \]
where \(m\) is the maximum of \(\left|\sum_{k=1}^{n}\varphi_k(\xi)\right|\). For the function \(\varphi(\xi)\) the corresponding quantity is
\[ \gamma(Ce\cap\sigma_z^r,\varphi)=\lim_{\xi\to\infty}\xi\varphi(\xi) =\frac{1}{1+m}\sum_{k=1}^{n}\gamma(Ce\cap\sigma_k,\varphi_k)\ge \]
\[ \ge \frac{1}{1+m}\sum_{k=1}^{n}\frac{1}{r}(r_k)^2 \ge \frac{C_2S}{\pi r(1+m)} . \]
Assuming that \(\xi\) does not belong to any of the disks \(\{\sigma_k'\}\), from Theorem 1 we obtain
\[ \left|\sum_{k=1}^{n}\varphi_k(\xi)\right| \le \sum_{k=1}^{n}|\varphi_k(\xi)| =\sum_{k=1}^{n}\left|\sum_{q=1}^{\infty}\frac{C_q^k}{(\xi-z_k)^2}\right|\le \]
\[ \le \sum_{k=1}^{n}\sum_{q=1}^{\infty}\max|\varphi_k(\xi)| \frac{A\gamma(Ce\cap\sigma_k)[d(Ce\cap\sigma_k)]^{q-1}}{|\xi-z_k|^q} \le \]
\[ \le \frac{C_3}{r}\,2\sqrt{\pi S}\le \frac{2C_3}{r}\sqrt{\pi r^2}=C_4, \]
where \(C_4\) is an absolute constant.
If, however, \(\xi\) belongs to some one of the disks \(\{\sigma_k'\}\), then
\[ \left|\sum_{k=1}^{n}\varphi_k(\xi)\right|\le 1+C_4. \]
Thus, \(m\le C_4+1\). Consequently,
\[ \gamma(Ce\cap\sigma_z^r)\ge \gamma(Ce\cap\sigma_z^r,\varphi)\ge \frac{C_5S}{r}, \]
i.e.
\[ P_z^r(e)\ge \frac{C_1S}{r^2}. \]
The lemma is proved.
Lemma 5. For an arbitrary set \(e\), at almost every one of its points of rarefaction (up to a set of planar measure \(0\)), the analytic density of the complement is positive.
We omit the proof of Lemma 5 because of its cumbersomeness.
Theorem 2. If the set \(e\) has no points of condensation, then for all \(r\) and \(z\)
\[ P_z^r(Ce)\ge \pi C_1 \]
(see Lemma 2).
It will be proved below that, under the assumptions of Theorem 2, \(P_z^r(Ce)\) turns out to be identically equal to \(1\).
Received
30 IX 1958