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Maximal operators on BMO and slices

Shahaboddin Shaabani

Abstract

We prove that the Hardy-Littlewood maximal operator is discontinuous on $\bmorn$ and maps $\vmorn$ to itself. A counterexample to boundedness of the strong and directional maximal operators on $\bmorn$ is given, and properties of slices of $\bmorn$ functions are discussed.

Maximal operators on BMO and slices

Abstract

We prove that the Hardy-Littlewood maximal operator is discontinuous on and maps to itself. A counterexample to boundedness of the strong and directional maximal operators on is given, and properties of slices of functions are discussed.
Paper Structure (4 sections, 12 theorems, 62 equations)

This paper contains 4 sections, 12 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Discontinuity of $M$ on ${BMO}(\mathbb{R}^n)$
  3. The Uncentered Hardy-Littlewood Maximal Operator on ${VMO}(\mathbb{R}^n)$
  4. Slices of BMO functions and unboundedness of directional and strong maximal operators

Key Result

Theorem 2.1

Let $f$ be a nonnegative function supported in $[0,1]$, $\|f\|_{{L}^{\infty}}\le1$ and $\|f\|_{{L}^1}>\log2$. Then, there exists a sequence of bounded functions $\{f_n\}$ converging to $f$ in ${BMO}(\mathbb{R})$ such that $\{Mf_n\}$ does not converge to $Mf$ in ${BMO}(\mathbb{R})$.

Theorems & Definitions (26)

  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • proof : Proof of Theorem \ref{['discontinuitytheorem']}
  • Corollary 2.5
  • Theorem 3.1
  • Lemma 3.2
  • ...and 16 more