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Fractional Maximal operator in hyperbolic spaces

Gonzalo Ibañez-Firnkorn, Emanuel Ramadori

TL;DR

This work extends fractional maximal operator theory to the hyperbolic setting where the measure exhibits non-doubling exponential growth. It introduces a local weight class $A_{p,q}^{loc}(\mathcal{H}^n)$ and decomposes the operator into local and far parts, using distributional estimates for $A_{r,\alpha}$ and level-set methods to establish weighted weak-type and, under a beta<gamma condition, strong-type bounds. The results are complemented by explicit weight constructions showing weak-type can hold without strong-type and that the local-weight framework can exceed the classical $A_{p,q}$ class. Collectively, the paper advances weighted inequalities for fractional maximal operators in hyperbolic spaces and clarifies how exponential volume growth shapes these bounds.

Abstract

In this article, we introduce the fractional maximal operator on the Hyperbolic space, a non-doubling measure space, and study the weighted boundedness. Motivated in the weighted boundedness of Hardy-Littlewood maximal studied by Antezana and Ombrosi in [1], we give conditions for the weak type and strong type estimate for fractional maximal. Also, we provide examples of weights for which the fractional maximal operator satisface weak type $(p,q)$ inequality but strong type $(p,q)$ inequality fails.

Fractional Maximal operator in hyperbolic spaces

TL;DR

This work extends fractional maximal operator theory to the hyperbolic setting where the measure exhibits non-doubling exponential growth. It introduces a local weight class and decomposes the operator into local and far parts, using distributional estimates for and level-set methods to establish weighted weak-type and, under a beta<gamma condition, strong-type bounds. The results are complemented by explicit weight constructions showing weak-type can hold without strong-type and that the local-weight framework can exceed the classical class. Collectively, the paper advances weighted inequalities for fractional maximal operators in hyperbolic spaces and clarifies how exponential volume growth shapes these bounds.

Abstract

In this article, we introduce the fractional maximal operator on the Hyperbolic space, a non-doubling measure space, and study the weighted boundedness. Motivated in the weighted boundedness of Hardy-Littlewood maximal studied by Antezana and Ombrosi in [1], we give conditions for the weak type and strong type estimate for fractional maximal. Also, we provide examples of weights for which the fractional maximal operator satisface weak type inequality but strong type inequality fails.
Paper Structure (5 sections, 4 theorems, 68 equations)

This paper contains 5 sections, 4 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Proof of main results
  3. Proof of Theorem \ref{['AcotM']}
  4. Proof of Proposition \ref{['PropEx']}
  5. Examples

Key Result

Theorem 1.1

Let $0<\alpha<n$, $1\leq p<\frac{n}{\alpha}$ and $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$ and $w$ be a weight. Suppose that Then, Furthermore, if $\beta<\gamma$ then for each fixed $\delta\geq 0$ we have In particular, if $\beta<\gamma$ then

Theorems & Definitions (9)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 1.3
  • Corollary 1.4
  • Lemma 2.1
  • proof
  • proof : Proof of Theorem \ref{['AcotM']}
  • proof
  • Example 3.1