Spaces of Type BLO on Non-homogeneous Metric Measure Spaces

TL;DR

This paper introduces a new function space called RBLO on non-homogeneous metric measure spaces, explores its properties, and demonstrates its relevance in the boundedness of certain integral operators.

Contribution

It defines the RBLO space in non-homogeneous settings, shows it is contained within RBMO, and characterizes it with new criteria, extending the theory of function spaces.

Findings

RBLO is a subset of RBMO in non-homogeneous spaces

Characterizations for RBLO are established

Boundedness of Calderón-Zygmund operators from L∞ to RBLO is proved

Abstract

Let $({\mathcal X}, d, \mu)$ be a metric measure space and satisfy the so-called upper doubling condition and the geometrically doubling condition. In this paper, the authors introduce the space ${\mathop\mathrm{RBLO}}(\mu)$ and prove that it is a subset of the known space ${\mathop\mathrm{RBMO}}(\mu)$ in this context. Moreover, the authors establish several useful characterizations for the space ${\mathop\mathrm{RBLO}}(\mu)$. As an application, the authors obtain the boundedness of the maximal Calder\'on-Zygmund operators from $L^\infty(\mu)$ to ${\mathop\mathrm{RBLO}}(\mu)$.