The multilinear fractional sparse operator theory II: refining weighted estimates via multilinear fractional sparse forms (2502.17300v4)

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 ${\mathcal A}{\eta,\mathcal{S},\tau,{\vec{r}},s'}^\mathbf{b,k,t}$, 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 $\mathscr{M}{\eta,\vec{r}}$ 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 Calder\'on--Zygmund operators and multilinear fractional rough singular integral operators. (2) Investigating sparse form type weighted Lebesgue $L^p(\omega)$ and weighted Sobolev $W^{s,p}(\omega)$ regularity estimates for solutions of fractional Laplacian equations with higher-order commutators.