Multilinear estimates for maximal rough singular integrals

TL;DR

This paper advances the multilinear harmonic analysis of rough maximal singular integrals by establishing $L^{p_1}\times\cdots\times L^{p_m}\to L^p$ bounds for maximal multilinear rough singular integrals with kernel $K(oldsymbol{y})=\frac{\Omega(\boldsymbol{y}')}{|\boldsymbol{y}|^{mn}}$, under the vanishing moment condition and $\Omega\in L^q(\mathbb{S}^{mn-1})$ for $q>1$. The authors implement a dyadic decomposition of the kernel and a wavelet-based refinement, reducing the problem to controlling a sharp part $\mathcal{L}_{\Omega}^{\sharp}$ and proving two key propositions that yield decay in the dyadic frequency index $\mu$, which then feeds into a multi-parameter interpolation framework to cover all admissible exponents. A central feature is the use of multilinear paraproducts, shifted operator techniques, and a multi-sublinear Marcinkiewicz interpolation to transfer endpoint bounds to the full range of $p_j$ and $p$, culminating in $L^{p_1}\times\cdots\times L^{p_m}\to L^p$ boundedness for $\frac{1}{m}<p<\infty$. As an application, the paper proves almost everywhere convergence for the associated doubly truncated multilinear singular integrals, strengthening the connection between boundedness theory and pointwise convergence in the multilinear, rough-kernel regime.

Abstract

In this work, we establish $L^{p_1}\times \cdots\times L^{p_m}\to L^p$ bounds for maximal multi-(sub)linear singular integrals associated with homogeneous kernels $\frac{Ω(\vec{\boldsymbol{y}}')}{|\vec{\boldsymbol{y}}|^{mn}}$ where $Ω$ is an $L^q$ function on the unit sphere $\mathbb{S}^{mn-1}$ with vanishing moment condition and $q>1$. As an application, we obtain almost everywhere convergence results for the associated doubly truncated multilinear singular integrals.

Figures (4)

• Figure 1: The region $\mathcal{H}^3(s)$
• Figure 2: The trilinear case $m=3$ : the range of $(\frac{1}{p_1},\frac{1}{p_2},\frac{1}{p_3})$
• Figure 3: $(a_{\nu+1},a_{\nu+1},a_{\nu+1})$ when $m=3$
• Figure 4: The trilinear case $m=3$ : $\mathcal{H}^3(a_{\nu})$ and $\mathcal{C}^3(a_{\nu+1})$

Theorems & Definitions (16)

• Theorem A
• Theorem B
• Theorem 1
• Theorem 2
• Lemma C
• Lemma D
• Lemma E
• Lemma F
• Proposition 3
• Proposition 4