The multilinear fractional bounded mean oscillation operator theory I: sparse domination, sparse $T1$ theorem, off-diagonal extrapolation, quantitative weighted estimate -- for generalized commutators

This paper introduces and studies a class of multilinear fractional bounded mean oscillation operators (denoted {\rm $m$-FBMOOs}) defined on ball-basis measure spaces $(X, μ, \mathcal{B})$. These operators serve as a generalization of canonical classes, such as the multilinear fractional maximal operators, the multilinear fractional Ahlfors-Beurling operators, the multilinear pseudo-differential operators with multi-parameter Hörmander symbol, and some multilinear operators admitting $\mathbb{V}$-valued $m$-linear fractional Dini-type Calderón-Zygmund kernel representation. Crucially, the definition utilized here, incorporating the notion of "bounded mean oscillation," provides greater generality compared to those in Karagulyan (2019) and Cao et al. (2023). Our investigation systematically examines the properties of these operators and their generalized commutators through the lens of modern harmonic analysis, focusing on two principal directions: 1.We establish Karagulyan-type sparse domination for the generalized commutators. Subsequently, by developing bespoke dyadic representation theorems pertinent to this setting, we prove a corresponding multilinear fractional sparse $T1$ theorem for these generalized commutators. 2.With sparse bounds established, we obtain weighted estimates in multiple complementary methods: (1) Under a novel class of multilinear fractional weights, we prove four types of weighted inequalities: sharp-type, Bloom-type, mixed weak-type, and local decay-type.(2) We develop a multilinear non-diagonal extrapolation framework for these weights, which transfers boundedness flexibly among weighted spaces and establishes corresponding vector-valued inequalities.