Hölder exponents of Green's functions of Cantor sets
For \(\tau \in (0,1/2)\), let \(E_{\tau }\) denote the Cantor ternary set with scaling factor \(\tau \). It is known that the Green function \(g\) of \(\mathbb{C}_{\infty }\backslash E_{\tau }\) with pole at \(\infty \) is Hölder continuous of order \(\alpha \) for certain \(\alpha \in (0,1/2]\). This paper obtains improved estimates for \(\alpha \) in terms of \(\tau \). In the classical case where \(\tau =1/3\), it is shown that \(g\) is Hölder continuous of order at least \(0.3413\). The previously known lower bound was \( 0.1898\). The `best' value for \(\alpha \) here remains unknown.
scientific article; zbMATH DE number 5305528