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 .'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 .'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 spaces, 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.

Chalmoukis, N., Daskalogiannis, V. (2023). Holomorphic semigroups and Sarason’s characterization of vanishing mean oscillation. REVISTA MATEMATICA IBEROAMERICANA, 39(1), 321-340 [10.4171/RMI/1346].

Holomorphic semigroups and Sarason’s characterization of vanishing mean oscillation

Chalmoukis, Nikolaos
;
2023

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 .'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 .'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 spaces, 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.
Articolo in rivista - Articolo scientifico
Bloch space; bounded mean oscillation; generalized Volterra operator; maximal space of strong continuity; Möbius invariant spaces; Semigroups of composition operators;
English
28-apr-2022
2023
39
1
321
340
open
Chalmoukis, N., Daskalogiannis, V. (2023). Holomorphic semigroups and Sarason’s characterization of vanishing mean oscillation. REVISTA MATEMATICA IBEROAMERICANA, 39(1), 321-340 [10.4171/RMI/1346].
File in questo prodotto:
File Dimensione Formato  
Chalmoukis-2023-Revista Matematica Iberoamericana-VoR.pdf

accesso aperto

Descrizione: [C., Daskalogiannis, RMI, 2022] Holomorphic semigroups and Sarason_s characterization of vanishing mean oscillation.pdf
Tipologia di allegato: Publisher’s Version (Version of Record, VoR)
Licenza: Creative Commons
Dimensione 526.55 kB
Formato Adobe PDF
526.55 kB Adobe PDF Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/401676
Citazioni
  • Scopus 1
  • ???jsp.display-item.citation.isi??? 1
Social impact