complex analysis - Analytic Capacity

For a compact set $K\subset\mathbb{C}$ the analytic capacity is defined as
$$\gamma(K)=\sup\{|f^\prime(\infty)|:f\in M_K\}$$
where $M_K$ is the set of bounded holomorphic functions on $\mathbb{C}\backslash K$ with $\|f\|_\infty\le 1$ and $f(\infty)=0$.
I have two questions.

• What is the intuition behind this definition?


• What does $f^\prime(\infty)$ mean? It doesn't seem to be $\lim_{z\rightarrow\infty}f^\prime(z)$ so I'm confused.

It would also help me if somebody could point out a nice expository lecture notes/article on this topic (preferrably available online).

Answer

Let me first answer your second question : $f'(\infty)$ is defined in the following way :
$$f'(\infty):=\lim_{z \rightarrow \infty} z(f(z)-f(\infty)).$$
It is indeed equal to the coefficient of $1/z$ in the Laurent expansion of $f$ near $\infty$.

Regarding your first question, analytic capacity was first introduced by Ahlfors in the 1940's to study a problem of Painlevé about finding a "geometric" characterization of removable compact subsets of the plane. Let $K$ be a compact set in the plane. We say that $K$ is removable (for bounded holomorphic functions) if every function holomorphic and bounded outside $K$ is constant. For example, every single point is removable, by Riemann's classical theorem on removable singularities of holomorphic functions. By the same argument, every finite set is removable, and in fact one can prove that every countable compact set is removable. Furthermore, every removable compact set is totally disconnected, and in particular has empty interior and connected complement. However, there are non-removable totally disconnected compact sets : an easy example is a Cantor subset of the real line of positive Lebesgue measure.

The relation between removability of compact sets and analytic capacity is the following :

It is not difficult to show that a compact set $K$ is removable if and only if its analytic capacity is zero. Hence Painlevé's problem is about finding a geomtric characterization of compact sets of zero analytic capacity. This problem turned out to be very difficult but is now considered solved (it took more than a hundred years to solve it...). For a survey of Painlevé's problem, see the very nice survey by Xavier Tolsa (1).

I also recommend the book by Garnett (2) and the book by Dudziak (3). Analytic capacity turned out to be a central tool in the theory of uniform approximation of holomorphic functions (this was discovered by Vitushkin in the 1960's). If you're interested in the applications of analytic capacity to these kind of problems, I recommend the book by Zalcman (4).

If you have any more questions, feel free to ask me. I'm currently finishing a PhD Thesis on analytic capacity... Analytic capacity is a very interesting (and quite difficult in my opinion) research topic, and there are a lot of open problems.

Hope this helps,
Malik

References :

(1) Painlevé's problem and analytic capacity. X. Tolsa

(2) Analytic capacity and measure. J. Garnett

(3) Vitushkin's conjecture for removable sets. J. Dudziak

(4) Analytic capacity and rational approximation. L. Zalcman

EDIT
For a survey of results related to analytic capacity, I also suggest you take a look at our paper http://arxiv.org/abs/1209.3326

See especially Section II about the preliminaries on analytic capacity.