Finitely randomized dyadic systems and BMO on metric measure spaces. (June 2015)

• Record Type: Journal Article
• Authors: Heikkinen, Toni; Korvenpää, Janne
• Is Part Of: Nonlinear analysis. Volume 120(2015)
• Page Start: 30
• Page End: 42
• Publication Date: 2015-06
• Subjects: Randomized dyadic system -- Bounded mean oscillation -- Dyadic BMO -- Dyadic cube -- Metric measure space; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2015.02.015
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We study the connection between BMO and dyadic BMO in metric measure spaces using finitely randomized dyadic systems, and give a Garnett–Jones type proof for a theorem of Uchiyama on a construction of certain BMO functions. We obtain a relation between the BMO norm of a suitable expectation over dyadic systems and the dyadic BMO norms of the original functions in different systems. The expectation is taken over only finitely randomized dyadic systems to overcome certain measurability questions. Applying our result, we derive Uchiyama's theorem from its dyadic counterpart, which we also prove.