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Asymptotically sharp reverse Hölder inequalities for flat Muckenhoupt weights

Ioannis Parissis, Ezequiel Rela

Abstract

We present reverse Hölder inequalities for Muckenhoupt weights in $\mathbb{R}^n$ with an asymptotically sharp behavior for flat weights, namely $A_\infty$ weights with Fujii-Wilson constant $(w)_{A_\infty}\to 1^+$. That is, the local integrability exponent in the reverse Hölder inequality blows up as the weight becomes nearly constant. This is expressed in a precise and explicit computation of the constants involved in the reverse Hölder inequality. The proofs avoid BMO methods and rely instead on precise covering arguments. Furthermore, in the one-dimensional case we prove sharp reverse Hölder inequalities for one-sided and two sided weights in the sense that both the integrability exponent as well as the multiplicative constant appearing in the estimate are best possible. We also prove sharp endpoint weak-type reverse Hölder inequalities and consider further extensions to general non-doubling measures and multiparameter weights.

Asymptotically sharp reverse Hölder inequalities for flat Muckenhoupt weights

Abstract

We present reverse Hölder inequalities for Muckenhoupt weights in with an asymptotically sharp behavior for flat weights, namely weights with Fujii-Wilson constant . That is, the local integrability exponent in the reverse Hölder inequality blows up as the weight becomes nearly constant. This is expressed in a precise and explicit computation of the constants involved in the reverse Hölder inequality. The proofs avoid BMO methods and rely instead on precise covering arguments. Furthermore, in the one-dimensional case we prove sharp reverse Hölder inequalities for one-sided and two sided weights in the sense that both the integrability exponent as well as the multiplicative constant appearing in the estimate are best possible. We also prove sharp endpoint weak-type reverse Hölder inequalities and consider further extensions to general non-doubling measures and multiparameter weights.

Paper Structure

This paper contains 16 sections, 20 theorems, 97 equations.

Table of Contents

  1. Introduction and main results
  2. Main definitions and background.
  3. Description of main results
  4. Some auxiliary results
  5. Flat Ainfty weights
  6. The main lemma
  7. Covering lemmas on the real line
  8. Sharp reverse Hölder inequalities in one dimension.
  9. Reverse Hölder inequalities for one-sided weights
  10. Two-sided reverse Hölder inequalities in one-dimension
  11. Sharp weak-type reverse Hölder inequalities at the endpoint
  12. Extensions to general measures
  13. Asymptotically sharp reverse Hölder inequalities in higher dimensions
  14. The proof of Theorem \ref{['t.inftyn']}
  15. Extensions to the multiparameter setting
  16. ...and 1 more sections

Key Result

Theorem 1.6

Let $w\in A_\infty$ with $(w)_{A_\infty}\leq\delta$ for some $\delta\geq 1$ and let $Q$ be a cube in $\mathbb R^n$. Then for all $1\leq r <1+\frac{1}{2^n(\delta-1)}$ we have the reverse Hölder inequality

Theorems & Definitions (37)

  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Remark 1.9
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • ...and 27 more