The capacity γ+

© Springer International Publishing Switzerland 2014. The capacity γ+ of a compact set E ⊂ C is γ+(E):= sup{μ(E): supp(μ) ⊂ E, ║Cμ║ L∞(C) ≤ 1}, and the capacity γ+ of an arbitrary set A ⊂ C is defined as γ+(A) = sup{γ+(E): E ⊂ A, E compact}. Notice that γ+ is defined like γ in (1.1) with the additional constraint that f has to coincide with Cμ, where μ is some positive Radon measure supported on E (observe that (Cμ)′(∞) = −μ(C) for any Radon measure μ). To be precise, there is another little difference: in (1.1) we asked ║f║ L∞(C\E) ≤ 1, while in (4.1) ║f║ L∞(ℂ) ≤ 1 (for f = Cμ). Trivially, we have γ+(E) ≤ γ(E). In Chapter 6 we will show that the converse inequality γ(E) ≤ c γ+(E) also holds. For the moment in this chapter we will study the capacity γ+. We will characterize it in terms of the Cauchy transform and also in terms of curvature, and finally we will exploit some (basic) techniques of potential theory to get further information on γ+.