Boundedness of Lusin-area and functions on localized BMO spaces over doubling metric measure spaces

Let X be a doubling metric measure space. If X has the δ-annular decay property for some δ∈(0,1], the authors then establish the boundedness of the Lusin-area function, which is defined via kernels modeled on the semigroup generated by the Schrödinger operator, from localized spaces BMOρ(X) to BLOρ(X) without invoking any regularity of considered kernels. The same is true for the function and unlike the Lusin-area function, in this case, X is not necessary to have the δ-annular decay property. Moreover, for any metric space, the authors introduce the weak geodesic property and the monotone geodesic property, which are proved to be respectively equivalent to the chain ball property of Buckley. Recall that Buckley proved that any length space has the chain ball property and, for any metric space equipped with a doubling measure, the chain ball property implies the δ-annular decay property for some δ∈(0,1]. Moreover, using some results on pointwise multipliers of bmo(R), the authors construct a counterexample to show that there exists a non-negative function which is in bmo(R), but not in blo(R); this further indicates that the above boundedness of the Lusin-area and functions even in Rd with the Lebesgue measure or the Heisenberg group also improves the existing results.