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 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 G of step r ≥ 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. Mathematics Subject Classification
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