A new type of bmo space for non-doubling measures

Abstract

Let \(\mu\) be a Radon measure on \({\mathbb {R}}^{d}\) which may be non-doubling and only satisfies \(\mu (Q(x,l))\le C_{0}l^{n}\) for all \(x\in {\mathbb {R}}^{d}\), \(l(Q)>0\), with some fixed constants \(C_{0}>0\) and \(n\in (0,d]\). We introduce a new type of \(bmo(\mu )\) space which looks bigger than the \(rbmo(\mu )\) space of Dachun Yang(JAMS, 2005). And its four equivalent norms are established by constructing some special types of auxiliary doubling cubes. Then we further obtain that this new \(rbmo(\mu )\) space actually coincides with the \(rbmo(\mu )\) space of Dachun Yang.