Frequency of Harmonic functions in Carnot groups and for operators of Baouendi type

TL;DR

This paper introduces a new Almgren frequency concept for harmonic functions on Carnot groups and Baouendi operators, establishing monotonicity formulas that link growth properties to the structure of solutions in subelliptic contexts.

Contribution

It develops a novel frequency notion adapted to sub-Laplacians on Carnot groups and derives new monotonicity formulas for harmonic functions and Baouendi-type operators.

Findings

New Almgren frequency notion for sub-Laplacian solutions

Monotonicity formulas linking frequency growth to solution structure

Deeper understanding of harmonic functions in subelliptic settings

Abstract

We introduce a new notion of Almgren's frequency which is adapted to solutions of a sub-Laplacian (harmonic functions) on a Carnot group of arbitrary step $\bG$. With this notion we investigate some new functionals associated with the frequency, and obtain monotonicity formulas for the relevant harmonic functions, or for the solutions of a closely connected class of degenerate second order operators of Baouendi type. The results proved in this paper provide some new insight into the deep link existing between the growth properties of the frequency, and the local and global structure of the relevant harmonic functions in these non-elliptic, or subelliptic, settings