Some estimates of multi-sublinear operators and commutators on mixed $\lambda$-central Morrey spaces

Abstract

In this paper, we establish the boundedness of multi-sublinear operators ${\mathcal {T}}_{m},$ ${\mathcal {T}}_{\alpha ,m},$ multi-sublinear commutators ${\mathcal {T}}^{{\vec {b}}}_{m},$ ${\mathcal {T}}^{{\vec {b}}}_{\alpha ,m}$ and iterated commutators ${\mathcal {T}}^{\Pi {\vec {b}}}_{m},$ ${\mathcal {T}}^{\Pi {\vec {b}}}_{\alpha ,m}$ generated with mixed $\lambda$-central ${\textrm{BMO}}$ functions ${\vec {b}}$ on mixed $\lambda$-central Morrey spaces ${\mathcal {B}}^{\vec {q},\lambda }({\mathbb {R}}^{n}),$ respectively, where ${\vec {b}}=(b_{1},\ldots ,b_{m}),$ $\vec {q}=(q_{1},\ldots ,q_{m})$ and $b_{i}\in {\textrm{CBMO}}^{\vec {q_{i}},\lambda _{i}}({\mathbb {R}}^{n})$ for $i=1,2,\ldots ,m.$ Similar results still hold for multilinear maximal operators $T^{*},$ their commutators $T^{*}_{{\vec {b}}}$ and $T^{*}_{\Pi {\vec {b}}}.$ In addition, we also derive the boundedness of multilinear commutators $I^{*}_{\alpha ,{\vec {b}}}$ of multilinear fractional maximal operators $I^{*}_{\alpha }$ on mixed $\lambda$-central Morrey spaces ${\mathcal {B}}^{\vec {q},\lambda }({\mathbb {R}}^{n}).$ As applications, we obtain the boundedness of the multilinear Calderón–Zygmund operators $T_{m},$ multilinear fractional operators $T_{\alpha ,m}$ and their commutators generated with mixed $\lambda$-central ${\textrm{BMO}}$ functions on mixed $\lambda$-central Morrey spaces ${\mathcal {B}}^{\vec {q},\lambda }({\mathbb {R}}^{n}),$ respectively.