Spaces of type BLO on non-homogeneous metric measure

Abstract

Let (X , d, μ) be a metric measure space satisfying the so-called upper doubling condition and the geometrically doubling condition. In this paper, we introduce the space RBLO(μ) and prove that it is a subset of the known space RBMO(μ) in this context. Moreover, we establish several useful characterizations for the space RBLO(μ). As an application, we obtain the boundedness of the maximal Calderón-Zygmund operators from L ∞ (μ) to RBLO(μ).

The Hardy Space H1 with Non-doubling Measures and Their Applications

“…Observe that, in .R D ; j j; dx/, u 2 A 1 .R D / implies that u satisfies the reverse Hölder inequality and hence u 2 L q loc .R D / for some q 2 .1; 1/ sufficiently close to 1, from which, it further follows that, for u 2 A 1 .R D /, p 2 .0; 1/ and all bounded functions f 1 and f 2 with bounded support, Moreover, the RBLO.X ; / norm of f is defined to be the minimal constant C as above and denoted by kf k RBLO .X ; / . In [87], Lin and Yang introduced the space RBLO.X ; /, which is a proper subset of RBMO .X ; /. Some equivalent characterizations of RBLO.X ; / and the boundedness of maximal Calderón-Zygmund operators from L 1 .X ; / to RBLO.X ; / were also established in [87].…”

A new characterization of regularized BMO spaces on non-homogeneous spaces and its applications

“…These properties are still true on non-homogeneous spaces in the sense of Hytönen; see [7,1]. Recently, Lin and Yang [13] introduced the space of regularized BLO, which is a subspace of RBMO(µ), and established several useful characterizations of this space.…”

Boundedness of certain commutators over non-homogeneous metric measure spaces

“…Sawano et al [33] presented an example showing that, if (X , d, μ) is not geometrically doubling, then Morrey spaces depend on the auxiliary parameters. More research on function spaces and the boundedness of various operators on metric measure spaces of non-homogeneous type can be found in [1,3,4,17,21,[24][25][26][27][28]. We refer the reader to the survey [45] and the monograph [46] for more developments on harmonic analysis in this setting.…”