The multilinear fractional sparse operator theory II: refining weighted estimates via multilinear fractional sparse forms

TL;DR

The paper develops a vector-valued, multilinear fractional sparse domination framework on spaces of homogeneous type, introducing a new maximal operator $\mathscr{M}_{\eta,\vec{r}}$ and weight class $A_{(\vec{p},q),(\vec{r},s)}$ to obtain norm equivalences with $(m+1)$-linear fractional sparse forms. It proves a domination principle that yields sharp, Bloom-type weighted bounds for generalized commutators of multilinear fractional Calderón–Zygmund and rough singular integral operators, and establishes sharp exponents $\Theta$ governing the weight growth. These results enable sparse-form type $L^p(w)$ and weighted Sobolev $W^{s,p}(w)$ regularity for fractional Laplacian equations with higher-order commutators, and extend to operator-valued and rough kernels within homogeneous-type settings. The framework thus provides quantitative, structurally robust tools for nonlocal harmonic analysis and PDEs with nonlocal interactions.

Abstract

This paper refines the main results from our previous study on sparse bounds of generalized commutators of multilinear fractional singular integral operators in \cite{CenSong2412}. The key improvements are: 1. We replace pointwise domination with the $(m+1)$-linear fractional sparse form ${\mathcal A}_{η,\mathcal{S},τ,{\vec{r}},s'}^\mathbf{b,k,t}$, advancing the vector-valued multilinear fractional sparse form domination principle, and relax conditions from multilinear weak type boundedness to multilinear locally weak type boundedness $W_{\vec{p}, q}(X)$. 2. We introduce a multilinear fractional $\vec{r}$-type maximal operator $\mathscr{M}_{η,\vec{r}}$ and develop a new class of weights $A_{(\vec{p},q),(\vec{r}, s)}(X)$ to characterize it, establishing norm equivalence with the sparse forms. 3. This norm equivalence provides sharp quantitative weighted estimates for $(m+1)$-linear fractional sparse form, removing exponent parameter limitations and achieving sharp operator norm bounds. 4. We demonstrate applications in two ways: (1) Providing sharp or Bloom type estimates for generalized commutators of multilinear fractional Calderón--Zygmund operators and multilinear fractional rough singular integral operators. (2) Investigating sparse form type weighted Lebesgue $L^p(ω)$ and weighted Sobolev $W^{s,p}(ω)$ regularity estimates for solutions of fractional Laplacian equations with higher-order commutators.

Figures (3)

• Figure 1: The relationships between weights
• Figure 2: Divergence for $k_i$: Operator norm $\lambda^2$ vs. reduced bound $\lambda^{2-\epsilon}$. The logarithmic y-axis shows faster growth of $\left\|\mathcal{A}_{1, \mathcal{S}, \{1\}, 2, 2}^{\mathbf{b}, (2), (1)}\right\|$.
• Figure 3: Divergence for $\varTheta = 2$: Operator norm $\delta^2$ vs. reduced bound $\delta^{2-\epsilon}$. The log y-axis shows faster growth of $\left\|\mathcal{A}_{0, \mathcal{S}, \{1\}, 2, \frac{4}{3}}^{\mathbf{b}, (2), (1)}\right\|$.

Theorems & Definitions (55)

• Definition 1.1
• Remark 1.2
• Proposition 1.3
• Lemma 1.4
• proof : Proof of Lemma \ref{['pre_1']}
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Proposition 2.5