On properties of multipliers of Cauchy transforms
For \(\alpha>0\), \({\mathcal F}_\alpha\) denotes the set of functions \(g\) on \(\Delta= \{z:| z|<1\}\) such that \(g(z)=\int_\Gamma(1-xz)^{-\alpha} d\mu(x)\) for some complex-valued Borel measure \(\mu\) on the unit circle \(\Gamma\). \(M_\alpha\), the set of multipliers of \({\mathcal F}_\alpha\), is the set of functions \(f\) on \(\Delta\) such that \(fg\) is in \({\mathcal F}_\alpha\) for every \(g\) in \({\mathcal F}_\alpha\). Similar definitions are given for \({\mathcal F}_0\) and \(M_0\). Norms are defined on these spaces, under which they become Banach spaces. The present paper continues the work of the authors and of their collaborators. Their main result gives a bound on the length of the arc \(f\circ\gamma\) for \(f\) in \(M_\alpha\), and \(\gamma\) a rectifiable arc in a Stolz angle. This length is at most \(C\| f\|_{M_\alpha}\), where \(C\) depends on the Stolz angle and the geometry of \(\gamma\). In particular if \(\alpha>0\) and \(\gamma\) is a chord in the unit disk of length \(\ell\) at most 2, then the length of \(f\circ\gamma\) is at most \((C_3/ \ell)^{\alpha+1} \| f\|_{M_\alpha}\), where \(C_3\) is an absolute constant. Another consequence is a bound on the area of the image of the Stolz angle. These results are shown to be sharp. For instance, for \(\alpha >0\) and \(p>1\), there is a function \(f\) in \(M_\alpha\) such that \(\int^1_0 | f'(p)|^p dp=\infty\). Finally, they show that strong restrictions apply for all derivatives \(f^{(n)}\) of analytic functions which are multipliers of \({\mathcal F}_\alpha\), \(\alpha>0\).
scientific article; zbMATH DE number 1286535