Properties of a frequency of Almgren type for harmonic functions in Carnot groups

Abstract

The celebrated frequency function of Almgren (Proceedings of Japan–United States Sem., Tokyo. North-Holland, Amsterdam, 1979), and its local and global properties, play a fundamental role in several questions in partial differential equations and geometric measure theory. In this paper we introduce a notion of Almgren’s frequency functional in any Carnot group \(\mathbb {G}\), and we analyze some local and global consequences of the boundedness of the frequency. Although our results are the counterpart of by now well-known classical ones, their proof is much more delicate and involved than their elliptic predecessors, and serious new obstructions arise. A central motivation for our study is the fundamental open question whether harmonic functions (i.e., solutions of a sub-Laplacian) in a Carnot group \(\mathbb {G}\) of step \(r\ge 2\) possess the strong unique continuation property (scup). Among the results in this paper, in Theorem 4.3 we show that a quantitative answer to such question is in fact equivalent to proving the local boundedness of the frequency.