The Besov–Bourgain–Morrey spaces on \({\mathbb {R}}^n\) have proved a bridge connecting Bourgain–Morrey spaces with amalgam-type spaces. In this article, we introduce Besov–Bourgain–Morrey spaces \({\mathcal {M}}{\dot{B}}^{p,\tau }_{q,r}(X)\) on the space of homogeneous type \((X,\rho ,\mu )\) in the sense of Coifman and Weiss satisfying the reverse doubling property. Various properties of these spaces, such as the nontriviality, the approximation property in terms of a family of conditional expectation operators, and the interpolation property derived via the Calderón product are investigated. We also establish the predual and the dual spaces of \({\mathcal {M}}{\dot{B}}^{p,\tau }_{q,r}(X)\) and, in the proof of the latter, we use the Fatou property of block spaces in the weak local topology of \(L^{q'}(X)\) and the density of simple functions (that is, finite linear combinations of characteristic functions of dyadic cubes) in \({\mathcal {M}}{\dot{B}}^{p,\tau }_{q,r}(X)\) . Moreover, we establish the equivalent integral representation of the norm of \({\mathcal {M}}{\dot{B}}^{p,\tau }_{q,r}(X)\) and further apply it to obtain the boundedness of the Hardy–Littlewood maximal operator, fractional integral operators, fractional maximal operators, Calderón–Zygmund operators, and commutators on \({\mathcal {M}}{\dot{B}}^{p,\tau }_{q,r}(X)\) .
