Functions on Localized BMO Spaces over Doubling Metric Measure Spaces

Let X be a doubling metric measure space. In this paper, the authors introduce the notions of Property (M̃) and Property (P ) of X , prove that Property (M̃) implies Property (P ) and give some equivalent characterizations of Property (M̃) and Property (P ). If X has Property (P ), 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 g∗ λ function and, moreover, unlike the Lusin-area function, in this case, X is not necessary to have Property (P ). These results are new even on R with the Lebesgue measure and the Heisenberg group, and apply in a wide range of settings, for instance, to the Schrödinger operator or the degenerate Schrödinger operator on R, or the sub-Laplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups. Moreover, via some results on pointwise multipliers of bmo(R), the authors construct a counterexample to show that there exists a nonnegative function which is in bmo(R), but not in blo(R).