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Compact $T(1)$ theorem à la Stein

Árpád Bényi, Guopeng Li, Tadahiro Oh, Rodolfo H. Torres

Abstract

We prove a compact $T(1)$ theorem, involving quantitative estimates, analogous to the quantitative classical $T(1)$ theorem due to Stein. We also discuss the $C_c^\infty$-to-$CMO$ mapping properties of non-compact Calderón-Zygmund operators.

Compact $T(1)$ theorem à la Stein

Abstract

We prove a compact theorem, involving quantitative estimates, analogous to the quantitative classical theorem due to Stein. We also discuss the -to- mapping properties of non-compact Calderón-Zygmund operators.
Paper Structure (10 sections, 19 theorems, 93 equations)

This paper contains 10 sections, 19 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Calderón-Zygmund operators, BMO, and CMO
  3. T(1) theorems and the main result
  4. Classical and compact T(1) theorems
  5. T(1) theorems à la Stein
  6. An application of Theorem 1
  7. Proof of Theorem \ref{['THM:1']}
  8. Mapping properties of Calderón-Zygmund operators into CMO
  9. Lack of compactness of convolution operators in Lebesgue spaces
  10. BMO, VMO, and CMO are not weakly sequentially complete

Key Result

Lemma 2.3

Let $f \in \textit{BMO}$. Then, $f \in \textit{CMO}$ if and only if for any cube $Q \subset \mathbb{R}^d$.

Theorems & Definitions (39)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Definition 3.1
  • Theorem A: $T(1)$ theorem
  • Definition 3.2
  • Theorem B: compact $T(1)$ theorem
  • Remark 3.3
  • Remark 3.4
  • Theorem C: $T(1)$ theorem à la Stein
  • ...and 29 more