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Estimates for maximal Fourier multiplier operators on $\Bbb R^2$ via square functions

Shuichi Sato

Abstract

We consider certain Littlewood-Paley square functions on $\Bbb R^2$ and prove sharp estimates for them, from which we can deduce $L^p$ boundedness of maximal functions defined by Fourier multipliers of Bochner-Riesz type on $\Bbb R^2$. This is a generalization of a result due to A. Carbery 1983.

Estimates for maximal Fourier multiplier operators on $\Bbb R^2$ via square functions

Abstract

We consider certain Littlewood-Paley square functions on and prove sharp estimates for them, from which we can deduce boundedness of maximal functions defined by Fourier multipliers of Bochner-Riesz type on . This is a generalization of a result due to A. Carbery 1983.

Paper Structure

This paper contains 6 sections, 28 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Maximal functions and Littlewood-Paley inequalities
  3. Proofs of Theorems \ref{['thm1.2']} and \ref{['thm1.3']} under $(B.1)$ with $\Psi'>0$
  4. Proof of Theorem \ref{['thm1.1']} for case $(B.1)$ with $\Psi'>0$
  5. Proofs of Theorems \ref{['thm1.1']}, \ref{['thm1.2']}, \ref{['thm1.3']} in full generality
  6. Proof of Theorem $\ref{['thm1.5+']}$

Key Result

Theorem 1.1

Suppose that $\psi"\neq 0$ on $I$. Let $\lambda>0$. Then, there exists a positive constant $C_\lambda$ such that

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Lemma 2.1
  • Lemma 2.2
  • ...and 33 more