The goal here is to develop a theory for the scale of Beurling spaces Ap, and the spaces Bq, in the context of unbounded Ahlfors regular sets (as a replacement for the Euclidean ambient). In Section 3.1 we establish some functional analytic properties of these spaces, while in Section 3.2 we study the duality between Ap and Bq spaces. In Section 3.3 we explore the action of the Hardy-Littlewood maximal operator, and its Ls-based version, on these scales of spaces, and prove some boundedness results.