Analytic capacity

In complex analysis, the analytic capacity of a compact subset K of the complex plane is a number that denotes 'how big' a bounded analytic function on C  K can become. Roughly speaking, γ(K) measures the size of the unit ball of the space of bounded analytic functions outside K. In complex analysis, the analytic capacity of a compact subset K of the complex plane is a number that denotes 'how big' a bounded analytic function on C  K can become. Roughly speaking, γ(K) measures the size of the unit ball of the space of bounded analytic functions outside K. It was first introduced by Ahlfors in the 1940s while studying the removability of singularities of bounded analytic functions. Let K ⊂ C be compact. Then its analytic capacity is defined to be Here, H ∞ ( U ) {displaystyle {mathcal {H}}^{infty }(U)} denotes the set of bounded analytic functions U → C, whenever U is an open subset of the complex plane. Further, Note that f ′ ( ∞ ) = g ′ ( 0 ) {displaystyle f'(infty )=g'(0)} , where g ( z ) = f ( 1 / z ) {displaystyle g(z)=f(1/z)} . However, usually f ′ ( ∞ ) ≠ lim z → ∞ f ′ ( z ) {displaystyle f'(infty ) eq lim _{z o infty }f'(z)} . If A ⊂ C is an arbitrary set, then we define The compact set K is called removable if, whenever Ω is an open set containing K, every function which is bounded and holomorphic on the set Ω  K has an analytic extension to all of Ω. By Riemann's theorem for removable singularities, every singleton is removable. This motivated Painlevé to pose a more general question in 1880: 'Which subsets of C are removable?' It is easy to see that K is removable if and only if γ(K) = 0. However, analytic capacity is a purely complex-analytic concept, and much more work needs to be done in order to obtain a more geometric characterization. For each compact K ⊂ C, there exists a unique extremal function, i.e. f ∈ H ∞ ( C ∖ K ) {displaystyle fin {mathcal {H}}^{infty }(mathbf {C} setminus K)} such that ‖ f ‖ ≤ 1 {displaystyle |f|leq 1} , f(∞) = 0 and f′(∞) = γ(K). This function is called the Ahlfors function of K. Its existence can be proved by using a normal family argument involving Montel's theorem.