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Composition of rough singular integral operators on rearrangement invariant Banach type spaces

Jiawei Tan, Qingying Xue

Abstract

Let $Ω$ be a homogeneous function of degree zero and enjoy the vanishing condition on the unit sphere $\mathbb{S}^{n-1}(n\geq 2)$. Let $T_Ω$ be the convolution singular integral operator with kernel ${Ω(x)}{|x|^{-n}}$. In this paper, when $Ω\in L^{\infty}(\mathbb {S}^{n-1})$, we consider the quantitative weighted bounds of the composite operators of $T_Ω$ on rearrangement invariant Banach function spaces. These spaces contain the classical Lorentz spaces and Orlicz spaces as special examples. Weighted boundedness of the composite operators on rearrangement invariant quasi-Banach spaces were also given.

Composition of rough singular integral operators on rearrangement invariant Banach type spaces

Abstract

Let be a homogeneous function of degree zero and enjoy the vanishing condition on the unit sphere . Let be the convolution singular integral operator with kernel . In this paper, when , we consider the quantitative weighted bounds of the composite operators of on rearrangement invariant Banach function spaces. These spaces contain the classical Lorentz spaces and Orlicz spaces as special examples. Weighted boundedness of the composite operators on rearrangement invariant quasi-Banach spaces were also given.
Paper Structure (8 sections, 9 theorems, 117 equations)

This paper contains 8 sections, 9 theorems, 117 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminary
  3. RIBFS and RIQBFS
  4. Young function and Orlicz maximal operators
  5. Sparse family
  6. Proofs of Theorems \ref{['thm1.1']}
  7. Proofs of Theorem \ref{['thm1.2']}
  8. Applications

Key Result

Theorem 1.1

Let $\Omega_{1}, \Omega_{2}$ be homogeneous of degree zero, have the vanishing moment (e.1) and $\Omega_{1}, \Omega_{2} \in L^{\infty}\left(\mathbb{S}^{n-1}\right)$. Let $1<r<\infty$ and $\mathbb{X}$ be a RIBFS with $1 <p_{\mathbb{X}}\leq q_{\mathbb{X}}< \infty$, then there exist $q,q_0 >1$ such tha

Theorems & Definitions (17)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Lemma 2.1: per
  • Lemma 2.2: hyt
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6: hu2
  • ...and 7 more