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Let ${\mathcal X}$ be a doubling metric measure space. If ${\mathcal X}$ hasthe $\delta$-annular decay property for some $\delta\in (0,\,1]$, the authorsthen establish the boundedness of the Lusin-area function, which is defined viakernels modeled on the semigroup generated by the Schr\"odinger operator, fromlocalized spaces ${\rm BMO}_\rho({\mathcal X})$ to ${\rm BLO}_\rho({\mathcalX})$ without invoking any regularity of considered kernels. The same is truefor the $g^\ast_\labda$ function and unlike the Lusin-area function, in thiscase, ${\mathcal X}$ is not necessary to have the $\delta$-annular decayproperty. Moreover, for any metric space, the authors introduce the weakgeodesic property and the monotone geodesic property, which are proved to berespectively equivalent to the chain ball property of Buckley. Recall thatBuckley proved that any length space has the chain ball property and, for anymetric space equipped with a doubling measure, the chain ball property impliesthe $\delta$-annular decay property for some $\delta\in (0,1]$. Moreover, usingsome results on pointwise multipliers of ${\rm bmo}({\mathbb R})$, the authorsconstruct a counterexample to show that there exists a nonnegative functionwhich is in ${\rm bmo}({\mathbb R})$, but not in ${\rm blo}({\mathbb R})$; thisfurther indicates that the above boundedness of the Lusin-area and$g^\ast_\lambda$ functions even in ${\mathbb R}^d$ with the Lebesgue measure orthe Heisenberg group also improves the existing results.

Boundedness of Lusin-area and $g_\lambda^\ast$ Functions on Localized BMO Spaces over Doubling Metric Measure Spaces