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Pointwise estimates for rough operators in a metric measure framework under some Ahlfors regularity conditions

Diego Chamorro, Anca-Nicoleta Marcoci, Liviu-Gabriel Marcoci

Abstract

We establish a new pointwise estimate for a class of rough operators in the setting of metric measure spaces endowed with a measure which is Ahlfors regular. This pointwise inequality can be divided in two steps: the first one relies in a subrepresentation formula that involves a modified Riesz potential and the upper gradient of the function considered and the second step gives a pointwise control of the Riesz potential in terms of a maximal function and a Morrey norm. We also investigate a family of functional inequalities that can be deduced from this pointwise estimate.

Pointwise estimates for rough operators in a metric measure framework under some Ahlfors regularity conditions

Abstract

We establish a new pointwise estimate for a class of rough operators in the setting of metric measure spaces endowed with a measure which is Ahlfors regular. This pointwise inequality can be divided in two steps: the first one relies in a subrepresentation formula that involves a modified Riesz potential and the upper gradient of the function considered and the second step gives a pointwise control of the Riesz potential in terms of a maximal function and a Morrey norm. We also investigate a family of functional inequalities that can be deduced from this pointwise estimate.
Paper Structure (4 sections, 3 theorems, 57 equations)

This paper contains 4 sections, 3 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Proof of the Theorem \ref{['Theo_RoughOpEstimate']}
  3. Proof of the Theorem \ref{['Theo_Pointwise_Morrey']}
  4. Functional inequalities

Key Result

Theorem 1

Consider $(X, d, \mu)$ a metric measure space endowed with a Ahlfors regular measure $\mu$ that satisfies the lower and upper condition (Ahlfors) with some power constant $0<\nu<+\infty$ and with constants $0<\mathfrak{c}_1\leq \mathfrak{c}_2<+\infty$. Assume also that the metric measure space $(X,d then for an operator $T^*_K$ defined in (MaximalOperator) associated to a kernel $K(\cdot, \cdot)$

Theorems & Definitions (3)

  • Theorem 1: Rough Operator estimate
  • Theorem 2: Morrey-type pointwise inequality
  • Theorem 3: A new pointwise inequality