Carleson Measures, Vanishing Mean Oscillation and Critical Points
Carlo Bellavita, Artur Nicolau, Georgios Stylogiannis
TL;DR
The paper addresses constructing boundary-regular outer functions in the quaternion-analytic class $\mathop{\mathrm{QA}}$ to convert Carleson measures into vanishing Carleson measures. It introduces a dyadic, stopping-time framework and Garnett–Jones $\mathrm{BMO}$ tools to produce an outer $E\in QA$ with $\log|E|\in\mathrm{VMO}$ (and in a variant, $\log|E|\in\mathrm{BMO}$) so that $|E|\mu$ is vanishing for any Carleson measure $\mu$, with sharpness results. The main consequences span Wolff’s theorem on boundary $\mathrm{VMO}$ behavior, derivative-factorization for Hardy-space functions yielding shared zeros with a $QA$-function, and compactness/non-boundedness properties of generalized Volterra operators when the symbol lies in $\mathrm{BMOA}$ or $\mathrm{VMOA}$. Sharpness results show the limits of the method: the vanishing condition cannot be strengthened in several senses, disk-algebra restriction is necessary, and $\log|E|$ cannot generally be improved from $\mathrm{BMO}$ to $\mathrm{VMO}$ in this context. The results provide a unified mechanism to control boundary regularity and critical-point behavior in Hardy spaces via outer-analytic constructions.
Abstract
Given a finite positive Borel measure $μ$ in the open unit disc of the complex plane, we construct a bounded outer function $E$ whose boundary values have vanishing mean oscillation such that $|E| μ$ is a vanishing Carleson measure. As an application it is shown that given any function in a Hardy space, there exists a bounded analytic function in the unit disc whose boundary values have vanishing mean oscillation, with the same critical points and multiplicities.