Holomorphic semigroups and Sarason's characterization of vanishing mean oscillation

TL;DR

This paper characterizes all semigroups of holomorphic functions that generalize Sarason's theorem on vanishing mean oscillation, showing they are elliptic and linking their properties to a logarithmic oscillation condition.

Contribution

It provides a complete characterization of semigroups replacing rotations in Sarason's theorem using a logarithmic vanishing oscillation condition, confirming they are elliptic.

Findings

Semigroups satisfying the logarithmic vanishing oscillation condition are elliptic.

The same class of semigroups applies to both BMOA and Bloch spaces.

The paper confirms the conjecture that all such semigroups are elliptic.

Abstract

It is a classical theorem of Sarason that an analytic function of bounded mean oscillation ($BMOA$), is of vanishing mean oscillation if and only if its rotations converge in norm to the original function as the angle of the rotation tends to zero. In a series of two papers Blasco et al. have raised the problem of characterizing all semigroups of holomorphic functions $(\varphi_t)$ that can replace the semigroup of rotations in Sarason's Theorem. We give a complete answer to this question, in terms of a logarithmic vanishing oscillation condition on the infinitesimal generator of the semigroup $(\varphi_t)$. In addition we confirm the conjecture of Blasco et al. that all such semigroups are elliptic. We also investigate the analogous question for the Bloch and the little Bloch space and surprisingly enough we find that the semigroups for which the Bloch version of Sarason's Theorem holds are exactly the same as in the $BMOA$ case.