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Weighted estimates for Multilinear Singular Integrals with Rough Kernels

Bae Jun Park

TL;DR

This work extends weighted norm inequalities to multilinear rough singular integrals with rough kernels by introducing a dyadic/Littlewood–Paley decomposition of the kernel and separating low- and high-frequency components. For the low-frequency part, standard multilinear Calderón–Zygmund theory with vector weights $A_{\vec{p}/q'}$ yields the desired bounds; the high-frequency part requires a novel sharp maximal bound that grows only polynomially in the frequency index, enabling summability through a multilinear Stein-type interpolation. The main result shows that, for $1<p_j<\infty$, $\frac{1}{m}<p<\infty$, $1<q\le\infty$, with $q'\le p_j$ and $\vec{w}\in A_{\vec{p}/q'}$, one has $\|\mathcal{L}_{\Omega}(f_1,\dots,f_m)\|_{L^{p}(v_{\vec{w}})} \lesssim \|\Omega\|_{L^q} \prod_j \|f_j\|_{L^{p_j}(w_j)}$. The analysis leverages maximal-function techniques, structural properties of multiple weights, and a complex interpolation framework adapted to multilinear operators with weights. The results advance the weighted theory for multilinear rough operators and provide quantitative bounds via $\|\Omega\|_{L^q}$ and the multiple-weight constants, with potential applications to further multilinear harmonic analysis problems.

Abstract

We establish weighted norm inequalities for a class of multilinear singular integral operators with rough kernels. Specifically, we consider the multilinear singular integral operator $\mathcal{L}_Ω$ associated with an integrable function $Ω$ on the unit sphere $\mathbb{S}^{mn-1}$ satisfying the vanishing mean condition. Extending the classical results of Watson and Duoandikoetxea to the multilinear setting, we prove that $\mathcal{L}_Ω$ is bounded from $L^{p_1}(w_1)\times\cdots\times L^{p_m}(w_m)$ to $L^p(v_{\vec{\boldsymbol{w}}})$ under the assumption that $Ω\in L^q(\mathbb{S}^{mn-1})$ and that the $m$ tuple of weights $\vec{\boldsymbol{w}}= (w_1,\ldots,w_m)$ lies in the multiple weight class $A_{\vec{\boldsymbol{p}}/q'}((\mathbb{R}^n)^m)$. Here, $q'$ denotes the Hölder conjugate of $q$, and we assume $q'\le p_1,\dots,p_m<\infty$ with $1/p = 1/p_1 + \cdots + 1/p_m$.

Weighted estimates for Multilinear Singular Integrals with Rough Kernels

TL;DR

This work extends weighted norm inequalities to multilinear rough singular integrals with rough kernels by introducing a dyadic/Littlewood–Paley decomposition of the kernel and separating low- and high-frequency components. For the low-frequency part, standard multilinear Calderón–Zygmund theory with vector weights yields the desired bounds; the high-frequency part requires a novel sharp maximal bound that grows only polynomially in the frequency index, enabling summability through a multilinear Stein-type interpolation. The main result shows that, for , , , with and , one has . The analysis leverages maximal-function techniques, structural properties of multiple weights, and a complex interpolation framework adapted to multilinear operators with weights. The results advance the weighted theory for multilinear rough operators and provide quantitative bounds via and the multiple-weight constants, with potential applications to further multilinear harmonic analysis problems.

Abstract

We establish weighted norm inequalities for a class of multilinear singular integral operators with rough kernels. Specifically, we consider the multilinear singular integral operator associated with an integrable function on the unit sphere satisfying the vanishing mean condition. Extending the classical results of Watson and Duoandikoetxea to the multilinear setting, we prove that is bounded from to under the assumption that and that the tuple of weights lies in the multiple weight class . Here, denotes the Hölder conjugate of , and we assume with .

Paper Structure

This paper contains 13 sections, 14 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Maximal inequalities
  4. Multiple weight classes
  5. Multilinear (convolution-type) Calderón-Zygmund operators
  6. Complex Interpolation for multilinear operators
  7. Proof of Theorem \ref{['singularthm']}
  8. The case when $\mu\le 0$
  9. The case when $\mu>0$
  10. Proof of Proposition \ref{['mainpropo']}
  11. Proof of \ref{['gq1l1myest']}
  12. Proof of \ref{['gq2l1myest']}
  13. Proof of \ref{['gq3l1myest']}

Key Result

Theorem A

Duan1993Wa1990 Let $\Omega$ be an integrable function on $\mathbb{S}^{n-1}$ with mean value zero. Let $1<p<\infty$ and $1<q\le \infty$ satisfy $q'\le p$. Suppose that $\Omega\in L^q(\mathbb{S}^{n-1})$ and $w\in A_{p/q'}(\mathbb R^n)$. Then there exists a constant $C$, depending on the $A_{p/q'}$-con for all $f\in\mathscr{C}_c^{\infty}(\mathbb R^n)\cap L^p(w)$.

Theorems & Definitions (20)

  • Theorem A
  • Theorem B
  • Definition C
  • Theorem 1
  • Lemma D
  • Lemma 2
  • proof
  • Lemma E
  • Lemma F
  • Lemma 3
  • ...and 10 more