Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Self-improving properties of weighted norm inequalities on metric measure spaces

Juha Kinnunen, Juha Lehrbäck, Antti V. Vähäkangas, Dachun Yang

Abstract

This work discusses self-improving properties of the Muckenhoupt condition and weighted norm inequalities for the Hardy-Littlewood maximal function on metric measure spaces with a doubling measure. Our main result provides direct proofs of these properties by applying a Whitney covering argument and a technique inspired by the Calderón-Zygmund decomposition. In particular, this approach does not rely on reverse Hölder inequalities.

Self-improving properties of weighted norm inequalities on metric measure spaces

Abstract

This work discusses self-improving properties of the Muckenhoupt condition and weighted norm inequalities for the Hardy-Littlewood maximal function on metric measure spaces with a doubling measure. Our main result provides direct proofs of these properties by applying a Whitney covering argument and a technique inspired by the Calderón-Zygmund decomposition. In particular, this approach does not rely on reverse Hölder inequalities.

Paper Structure

This paper contains 5 sections, 11 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Muckenhoupt weights
  4. Maximal function estimates
  5. Self-improvement results

Key Result

Lemma 3.5

Let $1<p<\infty$ and write $p'=\frac{p}{p-1}$. Then $w \in A_p$ if and only if $w^{1-p'} \in A_{p'}$.

Theorems & Definitions (22)

  • Definition 3.1
  • Definition 3.3
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • proof
  • Lemma 3.7
  • proof
  • Definition 4.1
  • Lemma 4.2
  • ...and 12 more