Holomorphic semigroups and Sarason's characterization of vanishing mean oscillation
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.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 arXiv:
 arXiv:2106.01294
 Bibcode:
 2021arXiv210601294C
 Keywords:

 Mathematics  Complex Variables;
 Mathematics  Functional Analysis;
 Primary: 30H05;
 47D06 47B33;
 Secondary 46E15
 EPrint:
 10 pages, 2 figures