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Sawyer estimates of mixed type for operators associated to a critical radius function

Fabio Berra, Gladis Pradolini, Pablo Quijano

Abstract

We prove mixed inequalities for the Hardy-Littlewood maximal function $M^{ρ,σ}$, where $ρ$ is a critical radius function and $σ\geq 0$. We also exhibit and prove an extension of Cruz-Uribe, Martell and Pérez extrapolation result in \cite{CruzUribe-Martell-Perez} to the setting of Muckenhoupt weights associated to a critical radius function $ρ$. This theorem allows us to give mixed inequalities for Schrödinger-Calderón-Zygmund operators, extending some previous estimates that we have already proved in \cite{BPQ}. Since we are dealing with unrelated weights, the proof involves a quite subtle argument related with the original ideas from Sawyer in \cite{Sawyer}.

Sawyer estimates of mixed type for operators associated to a critical radius function

Abstract

We prove mixed inequalities for the Hardy-Littlewood maximal function , where is a critical radius function and . We also exhibit and prove an extension of Cruz-Uribe, Martell and Pérez extrapolation result in \cite{CruzUribe-Martell-Perez} to the setting of Muckenhoupt weights associated to a critical radius function . This theorem allows us to give mixed inequalities for Schrödinger-Calderón-Zygmund operators, extending some previous estimates that we have already proved in \cite{BPQ}. Since we are dealing with unrelated weights, the proof involves a quite subtle argument related with the original ideas from Sawyer in \cite{Sawyer}.
Paper Structure (6 sections, 15 theorems, 114 equations)

This paper contains 6 sections, 15 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and definitions
  3. Proof of Theorem \ref{['thm: mixta para M']}
  4. Proof of Theorem \ref{['thm: extrapolacion']} and its consequences
  5. Application to Schrödinger type singular integrals
  6. Appendix

Key Result

Theorem 1.1

Let $\mathcal{F}$ be a family of pairs of functions satisfying that there exists $0<p_0<\infty$ such that the inequality holds for every $w\in A_\infty^\rho$, for every pair $(f,g)\in \mathcal{F}$ such that the left-hand side is finite and with $C$ depending only on $[w]_{A_\infty^\rho}$. Then the inequality holds for every $u\in A_1^\rho$ and $v\in A_\infty^\rho$.

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 3.1
  • Lemma 3.2: BPQ, Lemma 10
  • Lemma 3.3: Calderón-Zygmund decomposition on a cube
  • proof
  • Lemma 3.4
  • Lemma 3.5
  • proof : Proof of Theorem \ref{['thm: mixta para M']}
  • ...and 16 more