Recent work of Bui, Duong and Yan in [2] defined Besov spaces associated with a certain operator $L$ under the weak assumption that $L$ generates an analytic semigroup $e^{-tL}$ with Poisson kernel bounds on $L^2({\mathcal X})$ where ${\mathcal X}$ is a (possibly non-doubling) quasi-metric space of polynomial upper bound on volume growth. This note aims to extend certain results in [2] to a more general setting when the underlying space can have different dimensions at $0$ and infinity. For example, we make some extensions to the Besov norm equivalence result in Proposition 4.4 of [2], such as to more general class of functions with suitable decay at $0$ and infinity, and to non-integer $k\geq 1$.