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$L^p$-to-$L^q$ boundedness for commutators of the Cauchy transform

Adam Mair

Abstract

In this paper we prove a characterization of the $L^p$-to-$L^q$ boundedness of commutators to the Cauchy transform. Our work presents both new results and new proofs for established results. In particular, we show that the Campanato space characterizes boundedness of commutators for a certain range of $p$ and $q$.

$L^p$-to-$L^q$ boundedness for commutators of the Cauchy transform

Abstract

In this paper we prove a characterization of the -to- boundedness of commutators to the Cauchy transform. Our work presents both new results and new proofs for established results. In particular, we show that the Campanato space characterizes boundedness of commutators for a certain range of and .

Paper Structure

This paper contains 4 sections, 9 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['MainResult']}
  4. Sufficient Condition with p'=q

Key Result

Theorem 1.1

Let $b \in L^1_\text{loc}(\mathbb C)$, and suppose $p$ and $q$ satisfy $1 < p < 2 < q$, where $p' \neq q$. Given the quantity then $C_b:L^p(\mathbb C) \rightarrow L^q(\mathbb C)$ if, and only if,

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: Kolmogorov's Inequality
  • Definition 2.2
  • Theorem 2.3
  • Lemma 2.4
  • proof : Proof of Lemma \ref{['Fractional_linearcombo']}
  • Theorem 2.5: The Calderón-Zygmund Decomposition
  • Definition 2.6
  • Theorem 2.7: MR1291534
  • ...and 10 more