Aleksandrov operators as smoothing operators

Abstract

A holomorphic function $b$ mapping the unit disk $\text{disk}$ into itself induces a family of measures ${\tau }_{\alpha }$, $|\alpha |=1$, on the unit circle $\text{circle}$ by means of Herglotz's Theorem. This family of measures defines the Aleksandrov operator $A_b$ by means of the formula $A_{b}f(\alpha )={\int }_{\text{circle}}f(\zeta )d{\tau }_{\alpha }(\zeta )$, at least for continuous $f$. This operator preserves the smoothness classes determined by regular majorants, and is seen to be compact on these classes precisely when none of the measures ${\tau }_{\alpha }$ has an atomic part. In the process, a duality theorem for smoothness classes is proved, improving a result of Shields and Williams, and various theorems about composition operators on weighted Bergman spaces are extended to spaces arising from regular weights.