Analytic capacity

2020 Mathematics Subject Classification: Primary: 30C85 Secondary: 31A15 [MSN][ZBL]

Definitions

Analytic capacity was introduced by L.V. Ahlfors in [A] in 1947 for the characterization of removable singularities of bounded analytic functions. Let $K$ be a compact set in the complex plane $\mathbb C$. The analytic capacity of $K$ is defined by $$\gamma(K)=\sup\{\lim_{|z|\to\infty}|zf(z)|: f\in A(K)\}$$ where $A(K)$ is the set of functions which are analytic outside $K$, vanish at infinity and for which $|f(z)|\leq1$ for $z\in\mathbb C\setminus K$.

A related concept, which is more useful in rational approximation, is continuous analytic capacity $\alpha(K)$. It is defined as $\gamma(K)$ but the test functions $f$ are additionally required to be defined and continuous in the whole complex plane.

Removable sets

Ahlfors proved in [A] that a compact set $K$ is removable for bounded analytic functions if and only if $\gamma(K)=0$. The removability means that whenever $U$ is an open set in $\mathbb C$ containing $K$ and $f$ is bounded and analytic in $U\setminus K$, then $f$ has an analytic extension to $U$. It is fairly easy to show with the help of the Cauchy integral formula and Liouville's theorem that $K$ is removable for bounded analytic functions if and only if every bounded analytic function in $\mathbb C\setminus K$ is constant. Similarly, $\alpha(K)=0$ if and only if every bounded continuous function in $\mathbb C$ which is analytic in $\mathbb C\setminus K$ is constant.

An outstanding problem is to find a characterization in geometric terms for the null-sets of the analytic capacity. This is called Painlevè's problem since P. Painlevè studied it already in 1888 and proved a sufficient condition, property (i) below.

Properties

Some basic properties of the analytic capacity are the following three:

(i) If $K$ has length (that is, one-dimensional Hausdorff measure) zero, then $\gamma(K)=0$.

(ii) If $K$ has Hausdorff dimension greater than 1 (in particular, if $K$ has interior points), then $\gamma(K)>0$.

(iii) If $K$ is a subset of a rectifiable curve, then $\gamma(K)=0$ if and only if $K$ has length zero.

The first two are fairly easy, but the third one is deep, see, e.g., [P].

Rational approximation and semiadditivity

Solutions of many fundamental problems in rational approximation can be formulated in terms of analytic capacity. Thus any continuous function on a compact set $K$ in the plane can be uniformly approximated by rational functions with poles off $K$ if and only if $$\alpha(D\setminus K)=\alpha(D\setminus \text{interior}(K))\ \text{for any disc}\ D.$$ The work of the Moscow school (A.G. Vitushkin, M.S. Melnikov, and others) in the 1960's was particularly important in this development. Vitushkin also formulated the semiadditivity problem:

does there exist a constant $C$ such that for all compact subsets $K_1$ and $K_2$ of the plane, $$\gamma(K_1\cup K_2)\leq C(\gamma(K_1)+\gamma(K_2))?$$

Tolsa's solutions

In [T] X. Tolsa solved both Painlevè's problem and the semiadditivity problem. The solutions depend on the so-called Menger curvature $c(z_1,z_2,z_3)$ for triples of points in $\mathbb C$ and a formula of M.S. Melnikov relating it to the Cauchy kernel $1/z$. By definition the Menger curvature is the reciprocal of the radius of the circle passing through the points $z_1,z_2,z_3$; it is equal to zero if and only if the three points lie on one line. Tolsa proved the following two results:

Theorem 1 For a compact set $K\subset \mathbb C$, $\gamma(K)>0$ if and only there is a positive non-trivial Borel measure $\mu$ on $\mathbb C$ such that $\mu(D)\leq {\rm diam}\, (D)$ for all discs $D$ in $\mathbb C$ and $\int\int\int c(z_1,z_2,z_3)^2d\mu z_1d\mu z_2d\mu z_3<\infty$.

Theorem 2 The analytic capacity is semiadditive: there exists a constant $C$ such that for all compact subsets $K_1,K_2,\dots$, of the plane, $$\gamma(\cup_jK_j)\leq C\sum_j\gamma(K_j).$$

Good general references are [G], [P], [V] and [Z].

References

[A]	L.V. Ahlfors, "Bounded analytic functions" Duke Math. J. , 14 (1947) pp. 1–11. MR0021108Zbl 0030.03001
[G]	J.B. Garnett, "Analytic Capacity and Measure" Spinger-Verlag Lecture Notes 297, 1972. MR0454006 Zbl 0253.30014
[P]	H. Pajot, "Analytic Capacity, Rectifiability, Menger Curvature and the Cauchy Integral" , Spinger-Verlag Lecture Notes 1799, 2002. MR1952175 Zbl 1043.28002
[T]	X. Tolsa, "Painlevè's problem and the semiadditivity of analytic capacity" , Acta Mathematica 190 (2003), 105-149. MR1982794 Zbl 1060.30031
[V]	A.G. Vitushkin, "Analytic capacity of sets in problems of approximation theory"" Russian Math. Surveys , 22 : 6 (1967) pp. 139–200 Uspekhi Mat. Nauk , 22 : 6 (1967) pp. 141–199. MR1857292Zbl 0164.37701
[Z]	L. Zalcman, "Analytic capacity rational approximation" Spinger-Verlag Lecture Notes 50, 1968.