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

TL;DR

This paper derives a new pointwise estimate for rough operators in metric measure spaces with Ahlfors regular measures, linking Riesz potentials, maximal functions, and Morrey norms to establish functional inequalities.

Contribution

It introduces a novel pointwise inequality for rough operators in Ahlfors regular metric measure spaces, combining Riesz potentials and maximal functions.

Findings

Established a new pointwise estimate for rough operators.

Linked Riesz potentials with maximal functions and Morrey norms.

Derived functional inequalities from the pointwise estimate.

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.