The power 3/2 appearing in the estimate of analytic capacity
The analytic capacity of the compact plane set E is defined by \[ \gamma (E)=\sup_{f}| a_ 1|,\quad f(z)=a_ 0+a_ 1z^{-1}+...\quad at\quad \infty, \] where f runs through all functions analytic and bounded by 1 in \({\mathbb{C}}\setminus E\). This is a very subtle quantity and it has proved to be hard to relate to the geometric quantities. In ``A real variable method for the Cauchy transform, and analytic capacity'' (1988; Zbl 0645.30016) the author has proved that, if E lies on a rectifiable graph of length 1, then \(\gamma\) (E)\(\geq c(length P)^{3/2}\), P projection of E onto \({\mathbb{R}}\), where c is an absolute constant. In the pesent paper, the author shows that the exponent 3/2 is best possible. This is rather surprising in view of the fact that in some respects analytic capacity is related to linear measure. For the construction of his example, the author has to give careful estimates of the analytic capacity of particular sets using the Cauchy-Hilbert transform.
scientific article; zbMATH DE number 4198406