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Hi,

Here is another question that concerns analytic capacity. For a compact set $K$ in the plane, define the analytic capacity of $K$ by $$\gamma(K):=\sup|f'(\infty)|,$$ where the supremum is taken over all functions $f$ holomorphic and bounded by $1$ in the complement of $K$. Here

$$f'(∞)=\lim_{z \rightarrow \infty}z(f(z)−f(\infty)).$$

It was conjectured by Vitushkin, in view to applications in approximation theory, that analytic capacity is semi-additive, i.e. there exists a universal constant $C$ such that $$\gamma(E \cup F) \leq C(\gamma(E) + \gamma(F)).$$

This was proved by Tolsa in 2003. From what I know, wether or not analytic capacity is subadditive, i.e. if we can take $C=1$, is still open. However, I can't find much about this problem in the literature, hence the question :

What is the status on this problem? Has it been proved that analytic capacity is indeed subadditive in some particular cases?

The only thing I found about this is an article by Suita, "On subadditivity of analytic capacity for two continua.", in which it is proved that analytic capacity is subadditive in the case where $E,F$ are disjoint connected compact sets. The proof relies deeply on the fact that for compact connected sets, analytic capacity equals logarithmic capacity, which does not hold in the disconnected case.

Any reference about this is welcome.

Thank you, Malik

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