In this paper, the authors construct the Morrey spaces ${\mathcal{M}}_p^{\lambda }(\mathcal{V})$ on the metric measure spaces $(\mathcal{V},d,\mu )$ and explore their associated properties. Here, $(\mathcal{V},d,\mu )$ refers to metric measure spaces composed of infinite homogeneous trees $T$ equipped with the standard graph metric $d$ and a weighted measure $\mu$. As applications, the boundedness of the maximal operator and the fractional maximal operator related to admissible trapezoids on the Morrey spaces ${\mathcal{M}}_p^{\lambda }(\mathcal{V})$ is established.

Let $0<\alpha <n$ and $b$ be a locally integrable function. In this paper, we obtain the characterization of compactness of the iterated commutator $(T_{\mathrm{\Omega },\alpha }{)}_b^m$ generated by the function $b$ and the fractional integral operator with the homogeneous kernel $T_{\mathrm{\Omega },\alpha }$ on ball Banach function spaces. As applications, we derive the characterization of compactness via the commutator $(T_{\mathrm{\Omega },\alpha }{)}_b^m$ on weighted Lebesgue spaces, and further obtain a necessary and sufficient condition for the compactness of the iterated commutator $(T_{\alpha }{)}_b^m$ generated by the function $b$ and the fractional integral operator $T_{\alpha }$ on Morrey spaces. Moreover, we also show the necessary and sufficient condition for the compactness of the commutator $[b,T_{\alpha }]$ generated by the function $b$ and the fractional integral operator $T_{\alpha }$ on variable Lebesgue spaces and mixed Morrey spaces.

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.