Let \(0<q\le p\le r\le \infty \) and \(\tau \in (0,\infty ]\) . In this article, we introduce a local variant \(\mathcal {M}B_{q,r}^{p,\tau }\) of Besov–Bourgain–Morrey spaces \(\mathcal {M}\dot{B}_{q,r}^{p,\tau }\) , whose special case \(\tau =r\) was originally introduced by J. Bourgain and has proved to play an important role in the study related to the Strichartz estimate and some partial differential equations. These local spaces \(\mathcal {M}B_{q,r}^{p,\tau }\) include Bourgain–Lebesgue, local Morrey, and amalgam spaces as special cases. We find the sufficient and necessary conditions, respectively, for their nontriviality, for \(\mathcal {M}\dot{B}_{q,r}^{p,\tau }\) to be properly contained in \(\mathcal {M}B_{q,r}^{p,\tau }\) , and also for both the boundedness and the Fefferman–Stein vector-valued maximal inequality about the local Hardy–Littlewood maximal operator on \(\mathcal {M}B_{q,r}^{p,\tau }\) . Moreover, on \(\mathcal {M}B_{q,r}^{p,\tau }\) we study their diversity, their duality, and their interpolation in terms of Calderón products. Using the Calderón product and a new pointwise sparse domination, we obtain the boundedness of local fractional integrals from Calderón products to \(\mathcal {M}B_{q,r}^{p,\tau }\) . Moreover, we also obtain the boundedness of local Calderón–Zygmund operators on \(\mathcal {M}B_{q,r}^{p,\tau }\) .
