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Weighted boundedness for the maximal operator associated with matrices

Gonzalo Ibañez-Firnkorn

Abstract

In this paper we study the boundedness on $L^p(w)$ of the maximal operator $M_{A^{-1}}$, defined by $M_{A^{-1}}f(x)=Mf(A^{-1}x)$, that is, the maximal of Hardy-Littlewood composed with a invertible matrix $A$. We present two different results of boundedness and provide a characterization for a particular case of matrices. The main novelty lies in examples illustrating the difference between the class of weights with a matrix, $\mathcal{A}_{A,p}$, and the classical Muckenhoupt weight class, $\mathcal{A}_{p}$. Finally, we extend these results to the fractional framework, considering the fractional maximal operator $M_{α, A^{-1}}$.

Weighted boundedness for the maximal operator associated with matrices

Abstract

In this paper we study the boundedness on of the maximal operator , defined by , that is, the maximal of Hardy-Littlewood composed with a invertible matrix . We present two different results of boundedness and provide a characterization for a particular case of matrices. The main novelty lies in examples illustrating the difference between the class of weights with a matrix, , and the classical Muckenhoupt weight class, . Finally, we extend these results to the fractional framework, considering the fractional maximal operator .
Paper Structure (13 sections, 18 theorems, 46 equations)

This paper contains 13 sections, 18 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Young Function and Luxemburg norm.
  4. Properties of the class $\mathcal{A}_{A,p}$
  5. Proof of main results
  6. Proof of Theorem \ref{['Fuerte']}
  7. Proof of Theorem \ref{['FuerteRH']}
  8. Proof for matrix with finite order
  9. Examples
  10. Not all Muckenhoupt weight is a weight in $\mathcal{A}_{A,p}$
  11. Not all weight in $\mathcal{A}_{A,p}$ is a Muckenhoupt weight
  12. Subclass of Muckenhoupt class
  13. Fractional context

Key Result

Theorem 1.1

Let $A$ be a invertible matrix, $1<p<\infty$. Suppose that $\varphi$ is a Young function such that $\overline{\varphi}\in B_{p}$ and $w\in \mathcal{A}_{A,p,\varphi}$ then $M_{A^{-1}}$ is bounded on $L^p(w)$.

Theorems & Definitions (28)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Proposition 1.6
  • Proposition 1.7
  • Proposition 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 18 more