On removable singularities for the analytic Zygmund class (Q1916870)

The bounded complex function \(f\) belongs to the Zygmund class \(\Lambda_*\) if \(|f(z+h)+f(z-h)-2f(z)|\leq\text{const}|h|\) for \(z,h\in\mathbb{C}\). The compact set \(K\subset\mathbb{C}\) is called \(\Lambda_*\)-removable if every \(f\in \Lambda_*\) that is analytic in \(\mathbb{C}\backslash K\) is analytic in \(\mathbb{C}\). Let \(\psi(t)=t^2[\log(1/t)\log\log\log(1/t)]^{1/2}\). The authors show that the Hausdorff measure corresponding to \(\psi\) is the ``correct'' measure for the removability problem: If \(\Lambda_\psi(K)=0\) then \(K\) is \(\Lambda_*\)-removable, but there exists \(K_1\) with \(\Lambda_\psi(K_1)<\infty\) that is not removable. This however does not give a complete characterization: There exists \(K_2\) with \(\Lambda_\psi(K_2)=\infty\) that is removable; this can also be deduced from a result of Kaufman.