In this paper, we define the $B_{\sigma }$-type grand Morrey spaces and establish the extrapolation theorem on the $B_{\sigma }$-type grand Morrey spaces. In the process of proving the theorem, we find that the predual spaces of these spaces are $H_{\sigma }$-block spaces and obtain the boundedness of the Hardy–Littlewood maximal operator on the predual spaces. By the extrapolation theory, the boundedness of the Calderón–Zygmund operator and commutators on the nonhomogeneous $B_{\sigma }$-type grand Morrey space is also obtained. In particular, the classical bounded mean oscillation (BMO) spaces are characterized by establishing the John–Nirenberg inequality on the nonhomogeneous $B_{\sigma }$ type grand Morrey space.

Let $0<\alpha <mn$ and $0<r,q<\mathrm{\infty }$. In this paper, we obtain the boundedness of some multilinear operators by establishing pointwise inequalities and applying extrapolation methods on tent spaces $T_r^q({\mathbb{R}}_{+}^{n+1})$, where these multilinear operators include multilinear Hardy–Littlewood maximal operator $\mathcal{M}$, multilinear fractional maximal operator ${\mathcal{M}}_{\alpha }$, multilinear Calderón–Zygmund operator $\mathcal{T}$, and multilinear fractional integral operator ${\mathcal{I}}_{\alpha }$. Therefore, the results of Auscher and Prisuelos–Arribas [Math. Z. 286 (2017), 1575–1604] are extended to the general case.

By introducing variable Fofana's spaces (L^p(·),L^q)^α( ${ℝ}^n$)(1<p(·)<∞, 1≤q,α≤∞) and their pre-dual spaces, we discussed the properties of those spaces. Meanwhile, via applying hierarchical decomposition of function and real variable techniques, the necessary and sufficient conditions for boundedness of fractional integral operators and their commutators on variable Fofana's spaces are obtained.