Exceptional families of measures on Carnot groups

TL;DR

This paper investigates special measure families on Carnot groups with zero p-module, providing conditions for p-exceptionality of intrinsic Lipschitz surfaces and describing broad classes of such surfaces.

Contribution

It establishes necessary and sufficient conditions for p-exceptionality of intrinsic Lipschitz surfaces in Carnot groups and characterizes a wide class of these surfaces.

Findings

Identified conditions for p-exceptionality of measure families.

Described a broad class of p-exceptional intrinsic Lipschitz surfaces.

Provided a comprehensive framework for understanding measure families on Carnot groups.

Abstract

We study the families of measures on Carnot groups that have vanishing $p$-module, which we call $p$-exceptional families. We found necessary and sufficient condition for the family of intrinsic Lipschitz surfaces passing through a common point to be $p$-exceptional for $p\geq 1$. We described a wide class of $p$-exceptional intrinsic Lipschitz surfaces for $p\in(0,\infty)$.