The multilinear fractional sparse operator theory I: pointwise domination and weighted estimate
Xi Cen, Zichen Song
TL;DR
This work advances multilinear fractional sparse operator theory on spaces of homogeneous type by establishing a pointwise sparse domination principle for ${\mathscr B}$-valued operators and their higher-order commutators. It introduces higher-order multi-symbol sparse operators ${\mathcal A}_{\eta,\mathcal S,\tau}^{\mathbf b,\mathbf k,\mathbf t}$ to achieve quantitative weighted bounds, and develops three complementary weight frameworks $A_{\vec p,q}^*$, $W_{\vec p,q}^\infty$, and $H_{\vec p,q}^\infty$, including Bloom-type estimates via maximal and iterated weight methods, as well as endpoint estimates. The results unify and extend prior sparse domination and Bloom-type analyses for both linear and multilinear fractional operators, with applications to a broad class of multilinear fractional Calderón–Zygmund-type operators and their commutators. The methodology leverages dyadic lattices on spaces of homogeneous type, grand maximal truncations, and Calderón–Zygmund decompositions to derive sharp quantitative bounds, enabling robust weighted control in non-diagonal, fractional, and higher-order settings. Overall, the paper provides a comprehensive framework linking sparse domination, Bloom-type estimates, and endpoint behavior for complex multilinear fractional operators, with potential impact onAnalysis in non-Euclidean settings and related harmonic analysis applications.
Abstract
How to establish some specific quantitative weighted estimates for the generalized commutator of multilinear fractional singular integral operator $\mathcal{T}_η^{\bf b}$ is the focus of this paper, which is defined by $$\mathcal{T}_η^{\bf b}(\vec{f})(x):= \mathcal{T}_η\left((b_1(x) - b_1)^{β_1}f_1,\ldots,(b_m(x) - b_m)^{β_m}f_m\right)(x),$$ where $\mathcal{T}_η$ is a multilinear fractional singular integral operator, ${\bf b}:=({b_1}, \cdots ,{b_m})$ is a set of symbol functions, and $({β_1}, \cdots ,{β_m}) \in {\mathbb{N}_0^m}$. Pointwise dominating the aforementioned commutator leads us to consider a class of higher order multi-symbol multilinear fractional sparse operator ${\mathcal A}_{η,\mathcal{S},τ}^\mathbf{b,k,t}$ to achieve this long-cherished wish. Therefore, it suffices to construct its quantitative weighted estimates, which firstly include the characterization of several types of multilinear weighted conditions $A_{\vec p,q}^*$, $W_{\vec p,q}^\infty$, and $H_{\vec p,q}^\infty$. Within the scope of this work, Bloom type estimate for first order multi-symbol multilinear fractional sparse operator is established herein. Moreover, we derive two distinct Bloom type estimates for higher order multi-symbol multilinear fractional sparse operator by using "maximal weight method" and "iterated weight method" respectively, which not only refines some of Lerner's methods but greatly enhances the generality of our conclusions. Endpoint quantitative estimates for multilinear fractional singular integral operators and their first order commutators are also obtained as the last main result. It is also worthy of highlighting that some important multilinear fractional operators are applicable to our results as applications.