UDC 517.533.5

MATHEMATICS

M. S. MELNIKOV

ANALYTIC CAPACITY AND THE CAUCHY INTEGRAL

(Presented by Academician A. N. Kolmogorov, March 11, 1966)

The paper gives an estimate of the Cauchy integral over a contour in terms of the analytic capacity \(\gamma(E)\) \((^{2,3})\) of the set \(E\) of singular points of the integrand.

Theorem 1. Let \(E\) be a closed set in the unit disk \(|z|\leq 1\); let \(f(z)\) be a function analytic inside the disk outside \(E\), \(|f(z)|\leq 1\), and let \(f(z)\) be continuous on the unit circle. Then
\[ \left|\int_{|\zeta|=1} f(\zeta)\,d\zeta\right|\leq c\gamma(E), \]
where \(c\) is an absolute constant.

This problem, which arose in approximation theory, was posed by A. G. Vitushkin \((^{1}), \S 4\).

Here, with the aid of the estimate obtained, we prove the semiadditivity of analytic capacity for two sets separated by an analytic curve (Corollary 4).

Lemma 1. Let \(E\) be a bounded closed set; let \(f(z)\) be analytic outside \(E\); \(f(\infty)=0\); \(\operatorname{Im} f(z)\leq a\), \(a>0\). Then \(|f'(\infty)|\leq 2a\gamma(E)\).

The lemma is proved by the transformation
\[ \tilde f(z)=f(z)[f(z)-2ia]^{-1}. \]

Lemma 2. Let \(E\) be a bounded closed set; let the point \(a\) belong to the component of the complement of \(E\) containing infinity; let
\(\rho=\rho(a,E)\) be the distance from \(a\) to \(E\); let \(f(z)\) be analytic outside \(E\), \(|f(z)|\leq 1\). Then
\[ \rho[\rho-\gamma(E)]|f'(a)|\leq 2\gamma(E). \]

The lemma follows from the equality
\[ f'(a)=\frac{1}{2\pi i}\int_{\Gamma}\frac{f(\zeta)\,d\zeta}{(\zeta-a)^2} = \frac{1}{2\pi i}\int_{\Gamma} \left[ \frac{f(\zeta)-f(a)}{(\zeta-a)^2} - \frac{f(a)}{\zeta-a} \right]d\zeta, \]
where \(\Gamma\) is a closed rectifiable curve separating \(E\) from infinity and from the point \(a\), but not separating \(a\) from \(\infty\); moreover one may assume that \(\rho(a,\Gamma)\) is arbitrarily close to \(\rho\).

Remark. If one further assumes that \(f(a)=0\), then we obtain
\[ \rho[\rho-\gamma(E)]|f'(a)|\leq \gamma(E). \]
From this remark, by the transformation
\[ \varphi(z)=\tilde f(1/z), \]
one immediately derives

Lemma 3. Let a closed set \(E\) lie in the annulus \(r\leq |z|\leq 1\); let \(E^*\) be the set symmetric to \(E\) with respect to the unit circle. Then
\[ r^2\gamma(E^*)\leq 2\gamma(E). \]

Theorem 2. Let \(E\) be a closed set in the unit disk \(|z|\leq 1\); let \(0\in E\); let \(E^*\) be symmetric to \(E\) with respect to the unit circle. Then
\[ \gamma(E\cup E^*)\leq 10[\gamma(E)+\gamma(E^*)]. \]

Proof. Let \(E_\delta, E_\delta^*\) be the \(\delta\)-neighborhoods of the sets \(E\) and \(E^*\). Obviously, as \(\delta\to0\), \(\gamma(E_\delta)\to\gamma(E)\), \(\gamma(E_\delta^*)\to\gamma(E^*)\); therefore it is enough

to show that \(\gamma(E\cup E^*)\le 10[\gamma(E_\delta)+\gamma(E_\delta^*)]\) for every \(\delta>0\). To this end we shall prove that \(|f'(\infty)|\le 10[\gamma(E_\delta)+\gamma(E_\delta^*)]\) for any function \(f(z)\), analytic outside \(E\cup E^*\), \(|f(z)|\le 1\), \(f(\infty)=0\). We note that then \(f(z)\) may be assumed continuous in the whole plane and analytic outside \(E_\delta\cup E_\delta^*\). Let \(z^*=1/\overline z\).

Denote by \(\varphi(z)\) that one of the two functions \(f(z)+\overline{f(z^*)}\), \(f(z)-\overline{f(z^*)}\), for which the modulus of the derivative at infinity is not less than \(|f'(\infty)|\); then \(|\varphi(z)|\le 2\), \(\varphi(z)\) is also analytic outside \(E_\delta\cup E_\delta^*\) and continuous everywhere, while on the unit circle \(\varphi(z)\) is either purely real or purely imaginary, depending on the choice.

Consider the functions

\[ \varphi_1(z)= \begin{cases} \displaystyle \varphi(z)-\frac{1}{2\pi i}\int_{|\zeta|=1}\frac{\varphi(\zeta)\,d\zeta}{\zeta-z}, & \text{for } |z|<1,\\[1.2em] \displaystyle \frac{1}{2\pi i}\int_{|\zeta|=1}\frac{\varphi(\zeta)\,d\zeta}{\zeta-z}, & \text{for } |z|>1; \end{cases} \]

\[ \varphi_2(z)= \begin{cases} \displaystyle \varphi(z)-\frac{1}{2\pi i}\int_{|\zeta|=1}\frac{\varphi(\zeta)\,d\zeta}{\zeta-z}, & \text{for } |z|>1,\\[1.2em] \displaystyle \frac{1}{2\pi i}\int_{|\zeta|=1}\frac{\varphi(\zeta)\,d\zeta}{\zeta-z}, & \text{for } |z|<1. \end{cases} \]

It is easy to see that \(\varphi_1(z)\) and \(\varphi_2(z)\) can be extended by continuity at the points of the unit circle not belonging to \(E_\delta\cup E_\delta^*\); then \(\varphi_1(z)\) and \(\varphi_2(z)\) are analytic respectively outside \(E_\delta\) and \(E_\delta^*\), \(\varphi_1+\varphi_2=\varphi\), \(\varphi_1(\infty)=0\), \(|\varphi_2(\infty)|\le 1\).

It is well known that

\[ \left|\operatorname{Im}\left[\int_{|\zeta|=1}\frac{\tau(\zeta)\,d\zeta}{\zeta-z}\right]\right| \le 2\pi \max_{|\zeta|=1}|\tau(\zeta)| \]

for any real continuous function \(\tau(\zeta)\). Therefore, depending on the choice of \(\varphi(z)\), either \(\operatorname{Im}[\varphi_1(z)]\le 4\), \(\operatorname{Im}[\varphi_2(z)]\le 4\), or \(\operatorname{Re}[\varphi_1(z)]\le 4\), \(\operatorname{Re}[\varphi_2(z)]\le 4\); but then, by Lemma 1, we obtain

\[ |f'(\infty)|\le |\varphi'(\infty)|\le |\varphi_1'(\infty)|+|\varphi_2'(\infty)| \le 10[\gamma(E_\delta)+\gamma(E_\delta^*)]. \]

The theorem is proved.

Theorem 3. Let \(E\) be a closed set in the annulus \(r\le |z|\le 1\); let \(f(z)\) be a function analytic inside the disk outside \(E\), continuous in the whole disk; \(|f(z)|\le 1\). Then

\[ \left|\int_{|\zeta|=1} f(\zeta)\,d\zeta\right|\le c\,\frac{1}{r^2}\,\gamma(E), \]

where \(c\) is an absolute constant. It may be assumed that \(f(z)\) is continuous only on the unit circle.

Proof. Without loss of generality, one may assume \(f(z)\) smooth at the points of the unit circle not belonging to the set \(E\). If \(E\) separates \(0\) from infinity, then \(\gamma(E)\ge r\), and the assertion of the theorem is obvious. Consider the functions

\[ \varphi_{+,-}(z)= \begin{cases} \displaystyle f(z), & \text{for } |z|<1,\\ \displaystyle \pm \overline{f(1/\overline z)}, & \text{for } |z|>1, \end{cases} \]

then the functions

\[ \psi_{+,-}(z)=\varphi_{+,-}(z)-\frac{1}{2\pi i} \int_{|\zeta|=1}\frac{f(\zeta)\pm \overline{f(\zeta)}}{\zeta-z}\,d\zeta, \]

extended by continuity at the points of the unit circle not belonging to \(E\), will be analytic outside \(E\cup E^*\), and moreover (analogously to the theo-

rem 2) \(\operatorname{Re}[\psi_+(z)] \leqslant 3,\ \operatorname{Im}[\psi_-(z)] \leqslant 3\). Then, by Lemma 1 and Theorem 2, we have

\[ |\psi'_{+,-}(\infty)| \leqslant 6[\gamma(E\cup E^*)]\leqslant 60[\gamma(E)+\gamma(E^*)]. \]

Further,

\[ \left|\int_{|\zeta|=1} f(\zeta)d\zeta\right| \leqslant \frac12 |\psi'_+(\infty)+\psi'_-(\infty)| \leqslant 60[\gamma(E)+\gamma(E^*)], \]

whence, by Lemma 3, we obtain the assertion of the theorem.

Proof of Theorem 1. Without loss of generality, one may assume that the set \(E\) is bounded by a finite number of rectifiable closed curves, and that \(f(z)\) is analytic inside the unit disk outside \(E\) and is smooth everywhere. Denote by \(\beta(z)\) a real-valued infinitely differentiable function in the unit disk with the properties: \(\beta(z)=1\) for \(1\geqslant |z|\geqslant {}^2/{}_3\); \(\beta(z)=0\) for \(|z|\leqslant {}^1/{}_3\); \(|\beta(z)|\leqslant 1,\ |\partial\beta/\partial\bar z|\leqslant 4\). Then

\[ \beta f+(1-\beta)f=f. \]

By the Cauchy–Green formula, one may represent \(\beta f=f_1+\varphi_1,\ (1-\beta)f=f_2+\varphi_2\), where \(f_1(z)\) is analytic in the disk outside that part of \(E\) which lies in the annulus \({}^1/{}_3\leqslant |z|\leqslant 1\); \(f_2(z)\) is analytic in the disk outside that part of \(E\) which lies in the disk \(|z|\leqslant {}^2/{}_3\); \(|f_1(z)|\leqslant c_1;\ |f_2(z)|\leqslant c_1;\ \varphi_1(z)+\varphi_2(z)=0\). Then

\[ \left|\int_{|\zeta|=1} f(\zeta)d\zeta\right| \leqslant \left|\int_{|\zeta|=1} f_1(\zeta)d\zeta\right| + \left|\int_{|\zeta|=1} f_2(\zeta)d\zeta\right|, \]

and the assertion of the theorem follows at once from Theorem 3 (with \(r={}^1/{}_3\)) and the obvious fact that the estimate of the integral holds if the set lies in the disk \(|z|\leqslant {}^2/{}_3\).

Corollary 1. If the sets \(E_i\) lie, respectively, in nonintersecting disks \(K_i\), then

\[ \gamma\left(\bigcup_i E_i\right)\leqslant c\sum_i \gamma(E_i). \]

Corollary 2. Let \(E\) be a closed set situated in the annulus \({}^1/{}_2\leqslant |z|\leqslant 1\); \(f(z)\) a continuous function, analytic in this annulus outside the set \(E\); \(|f(z)|\leqslant 1\). Then

\[ \left| \int_{|\zeta|=1} f(\zeta)d\zeta - \int_{|\zeta|={}^1/{}_2} f(\zeta)d\zeta \right| \leqslant c\gamma(E). \]

This corollary can be obtained in the same way as the proof of Theorem 1. Corollary 2 makes it possible to prove

Corollary 3. Let \(E\) be a bounded closed set; let \(a\in E\); \(K(n)\) be the annulus \({}^1/{}_{2^n}\geqslant |z-a|\geqslant {}^1/{}_{2^{n+1}}\). In order that there exist a function \(f(z)\), admitting uniform approximation on \(E\) by rational fractions and such that \(f(a)=1,\ f(z)<1\) for \(z\ne a\), it is necessary and sufficient that the series

\[ \sum_{n=0}^{\infty} 2^n\gamma[K(n)\setminus E] \]

converge.

Theorem 4. Let the domain \(G\) be bounded by a simple closed analytic curve \(\Gamma\); let the closed set \(E\subset \overline{G}\), and let \(f(z)\) be a continuous function, analytic in \(G\setminus E\); \(|f(z)|\leqslant 1\). Then

\[ \left|\int_{\Gamma} f(\zeta)d\zeta\right|\leqslant c(G)\gamma(E), \]

where \(c(G)\) is a constant depending only on the domain \(G\).

Without loss of generality, one may assume that \(E\subset G\) and that its boundary \(\partial E\) consists of a finite number of closed rectifiable curves. In this situation, for the proof of Theorem 4 we shall use dual formulations for the extremal problems arising in the definition of ana-

analytic capacity and of the maximum of the Cauchy integral (see, for example, \((^4)\)). Let \(A(E)\) be the set of functions \(f(z)\), analytic in \(CE\), such that \(|f(z)| \leq 1\), \(f(\infty)=0\); let \(H_1(E)\) be the space of type \(H_1\) of functions analytic in \(CE\) and equal to 0 at infinity; let \(A(G\setminus E)\) be the set of functions analytic in \(G\setminus E\), \(|f(z)| \leq 1\); and let \(H_1(G\setminus E)\) be the space of type \(H_1\) of functions analytic in \(G\setminus E\). Then

\[ \gamma(E)=\max_{f\in A(E)}\left|\int_{\partial E} f(\zeta)\,d\zeta\right| =\min_{g\in H_1(E)}\int_{\partial E}|1-g|\,ds, \]

\[ I(G,E)=\max_{f\in A(G\setminus E)}\left|\int_{\Gamma} f(\zeta)\,d\zeta\right| =\min_{g\in H_1(G\setminus E)} \left[\int_{\Gamma}|1-g|\,ds+\int_{\partial E}|g|\,ds\right]. \]

If \(G\) satisfies the condition of the theorem, then there exists a conformal mapping \(w(z)\) of the domain \(G\) onto the unit disk \(K\), which extends to a conformal mapping of some fixed neighborhood \(U(G)\) onto a neighborhood \(U(K)\), and moreover

\[ 0<m\leq |w'(z)|\leq M. \]

Let \(E_1\subset K\) be the image of \(E\) under the mapping \(w(z)\), and let \(g(w)\in H_1(K\setminus E_1)\) be a function such that

\[ I(K,E_1)=\int_{|w|=1}|1-g(w)|\,ds+\int_{\partial E_1}|g(w)|\,ds. \]

Then

\[ I(G,E)\leq \int_{\Gamma}|1-g(w(z))|\,ds+\int_{\partial E}|g(w(z))|\,ds \leq \frac{1}{m} I(K,E_1)\leq \frac{1}{m}c\gamma(E_1). \]

It remains to estimate \(\gamma(E_1)\) in terms of \(\gamma(E)\); this is done with the aid of the inverse mapping \(z(w)\), on the basis of the fact that

\[ \gamma(E)\geq c(G)I(U(G),E)\geq c(G)\frac{1}{M}I(U(K),E_1)\geq \frac{1}{M}c(G)\gamma(E_1). \]

Corollary 4. If two sets \(E_1\) and \(E_2\) are separated by an analytic curve \(\Gamma\), then

\[ \gamma(E_1\cup E_2)\leq c(\Gamma)\,[\gamma(E_1)+\gamma(E_2)]. \]

Theorem 5. If a bounded closed set \(E\) is such that its interior boundary (i.e., boundary points which are not boundary points for components of the complement) lies on an analytic curve, then every function continuous on \(E\) and analytic at the interior points of \(E\) can be uniformly approximated on \(E\) by rational functions of \(z\).

The proof follows from Theorems 3 and 7 of \((^5)\) and Theorem 4.

The author expresses his deep gratitude to A. G. Vitushkin for his attention to the work and for valuable advice.

Moscow State University
named after M. V. Lomonosov

Received
11 III 1966

REFERENCES

\(^1\) A. G. Vitushkin, Izv. AN SSSR, ser. matem., 28, 4 (1964).
\(^2\) L. V. Ahlfors, Duke Math. J., 14, 1 (1947).
\(^3\) A. G. Vitushkin, DAN, 123, No. 5 (1958).
\(^4\) S. Ya. Khavinson, DAN, 88, No. 6 (1953).
\(^5\) A. G. Vitushkin, DAN, 171, No. 6 (1966).