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On the analytic capacity \(\gamma_ +\). (English) Zbl 1041.31002
The analytic capacity \(\gamma_+\) of a compact set \(E\subset\mathbb{C}\) is defined by \(\gamma_+(E)= \sup_\mu\,\mu(E)\), where \(\mu\) is a positive Radon measure supported on \(E\) such that the Cauchy transform \(f(1/z)*\mu\) is a function in \(L^\infty(\mathbb{C})\) with \(\| f\|_\infty\leq 1\). \(\gamma_+\) was first introduced by T. Murai [A real variable method for the Cauchy transform, and analytic capacity (Lecture Notes in Mathematics 1307, Springer-Verlag, Berlin) (1988; Zbl 0645.30016)] to study the Cauchy transform. In general \(\gamma_+(E)\leq \gamma(E)\), where \(\gamma\) is the ‘usual’ analytic capacity of \(E\).
Let \(M_+(\mathbb{C})\) be the set of all positive finite Radon measures \(\mu\) on \(\mathbb{C}\), and \(M\) the maximal radial operator \(M\mu(x)= \sup_{r>0}\, \mu(B(x, r))/r\). Also, for any three pairwise distinct points \(x\), \(y\) and \(z\) in \(\mathbb{C}\), their Menger curvature is \(c(x,y,z)= 1/R(x,y,z)\), where \(R(x,y,z)\) is the radius of the circle through \(x\), \(y\) and \(z\); then for \(\mu\in M_+(\mathbb{C})\) let \[ c_\mu(x)= \biggl(\iint c(x,y,z\biggr)^2 d\mu(y)\,d\mu(z))^{1/2}. \] Finally, let \(U_\mu(x)= M\mu(x)+ c_\mu(x)\).
There is a close connection between \(\gamma_+\) and the \(L^2\) and weak \((1,1)\) boundedness of the Cauchy transform. The author shows that although \(\gamma\) is a capacity not originated by a positive symmetric kernel it satisfies some properties analogous to such capacities; for instance, \(\gamma_+\) can be described in terms of a potential, and it admits a dual characterization involving \(U_\mu\) in terms of this potential. It follows that \(\gamma_+(E)= 0\) if and only if there exists some finite Radon measure \(\mu\) such that \(U_\mu(x)= \infty\) for all \(x\in E\). The author is also able to estimate the capacity \(\gamma_+\) of some Cantor sets.
Next, the author shows that \(U_\mu\) satisfies a minimum principle which implies that \(\gamma_+(E)\approx \gamma_+(\partial_{\text{out}} E)\), where \(\partial_{\text{out}}E\) is the outer boundary of \(E\) (that is, the intersection of \(E\) with the closure of the unbounded component of the complement of \(E\)), and obtains some density estimates involving \(\gamma_+\) related to instability.

31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
30C85 Capacity and harmonic measure in the complex plane
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