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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.

##### MSC:
 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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