Remarks about Besicovitch Covering Property in Carnot Groups of Step 3 and Higher

Besicovitch Covering Property for Homogeneous Distances on the Heisenberg Groups

We prove that the Besicovitch Covering Property (BCP) holds for homogeneous distances on the Heisenberg groups whose unit ball centered at the origin coincides with an Euclidean ball. We provide therefore the first examples of homogeneous distances that satisfy BCP in these groups. Indeed, commonly used homogeneous distances, such as (Cygan-)Korányi and Carnot-Carathéodory distances, are known ...

Isodiametric Inequality in Carnot Groups

The classical isodiametric inequality in the Euclidean space says that balls maximize the volume among all sets with a given diameter. We consider in this paper the case of Carnot groups. We prove that for any Carnot group equipped with a Haar measure one can find a homogeneous distance for which this fails to hold. We also consider Carnot-Carathéodory distances and prove that this also fails f...

A Ratio Ergodic Theorem for Multiparameter Non-singular Actions

We prove a ratio ergodic theorem for non-singular free Z and R actions, along balls in an arbitrary norm. Using a Chacon-Ornstein type lemma the proof is reduced to a statement about the amount of mass of a probability measure that can concentrate on (thickened) boundaries of balls in R. The proof relies on geometric properties of norms, including the Besicovitch covering lemma and the fact tha...

Schur-Pair Property and the Structure of Varietal Covering Groups

C-Class Functions and Remarks on Fixed Points of Weakly Compatible Mappings in G-Metric Spaces Satisfying Common Limit Range Property

In this paper, using the contexts of C-class functions and common limitrange property, common fixed point result for some operator are obtained.Our results generalize several results in the existing literature. Some examplesare given to illustrate the usability of our approach.