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Off-diagonal bloom weighted estimates for bilinear commutators

Yunan Zeng

Abstract

We prove the off-diagonal estimates of the bilinear iterated commutators in the two-weight setting. The upper bound is established via sparse domination, and the lower bound is proved by the median method. Our methods are so flexible so that it can be easily extended to the multilinear scenario.

Off-diagonal bloom weighted estimates for bilinear commutators

Abstract

We prove the off-diagonal estimates of the bilinear iterated commutators in the two-weight setting. The upper bound is established via sparse domination, and the lower bound is proved by the median method. Our methods are so flexible so that it can be easily extended to the multilinear scenario.

Paper Structure

This paper contains 10 sections, 15 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dyadic cube system
  4. Basic notations
  5. Weight
  6. Bilinear Calderón-Zygmund operators
  7. A sparse domination principle
  8. Bilinear locally weak $L^{\vec{r}}$-bounded
  9. Weighted estimates of the sparse form
  10. proof of the main theorem

Key Result

Theorem 1.1

Let $1\leqslant r_1 <p_1,q_1< \infty, 1\leqslant r_2 <p_2 <\infty, 1<p,q<s\leqslant \infty$, satisfy ${1}/{p}={1}/{p_1}+{1}/{p_2}, {1}/{q}={1}/{q_1}+{1}/{p_2},$ and $k_1\in \mathbb{N}$. Let $T$ be a bilinear operator and $b\in L_{\rm{loc}}^1(\mathbb{R}^n)$. Suppose that $T$ and $\mathcal{M}^{\#}_{T, and $\nu_1\in A_{\infty}$. Then:

Theorems & Definitions (25)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Theorem 4.1
  • proof
  • ...and 15 more