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Weak Factorizations of the Hardy Spaces in Terms of Multilinear Calderón-Zygmund Operators on Ball Banach Function Spaces

Yichun Zhao, Xiangxing Tao, Jiang Zhou

Abstract

In this paper, our main purpose is to establish a weak factorization of the classical Hardy spaces in terms of a multilinear Calderón-Zygmund operator on the ball Banach function spaces. Furthermore, a new characterization of the BMO space via the boundedness of the commutator generated by the multilinear Calderón-Zygmund operator is also obtained. The results obtained in this paper have generality. As examples, we apply the above results to weighted Lebesgue space, variable Lebesgue space, Herz space, mixed-norm Lebesgue space, Lorentz space and so on.

Weak Factorizations of the Hardy Spaces in Terms of Multilinear Calderón-Zygmund Operators on Ball Banach Function Spaces

Abstract

In this paper, our main purpose is to establish a weak factorization of the classical Hardy spaces in terms of a multilinear Calderón-Zygmund operator on the ball Banach function spaces. Furthermore, a new characterization of the BMO space via the boundedness of the commutator generated by the multilinear Calderón-Zygmund operator is also obtained. The results obtained in this paper have generality. As examples, we apply the above results to weighted Lebesgue space, variable Lebesgue space, Herz space, mixed-norm Lebesgue space, Lorentz space and so on.

Paper Structure

This paper contains 13 sections, 16 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Statement of Main Results
  3. Ball Banach Function Spaces
  4. Multilinear Calderón-Zygmund operators
  5. The Atomic Hardy Spaces $H^1(\mathbb R^n)$
  6. Main Results
  7. The proofs of Theorem \ref{['the factorization']} and Theorem \ref{['the characterization BMO']}
  8. Some Examples
  9. Weighted Lebesgue Spaces
  10. Variable Lebesgue Spaces
  11. Herz Spaces
  12. Mixed-norm Lebesgue Spaces
  13. Lorentz Spaces

Key Result

Theorem 2.1

Let $1 \leqslant l \leqslant m$, $1<p_i<\infty$ for all $i=1, 2, \dots, m$ and $1/p_0=1/p_1+1/p_2+\dots+1/p_m$ with $1\leqslant p_0<\infty$. Assume the function $b\in \rm BMO$, the operator $T$ and commutators $[b,T]_l$ are bounded from $X^{p_1}\times \dots \times X^{p_m}$ to $X^{p_0}$. Then, for an Moreover, the following equivalence of norm holds

Theorems & Definitions (30)

  • Definition 2.1
  • Remark 2.1
  • Definition 2.2
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 3.1
  • ...and 20 more