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 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

Carleman estimates for Baouendi–Grushin operators with applications to quantitative uniqueness and strong unique continuation

“…We mention, in this connection, the remarkable fact that even the weak unique continuation property fails for −∆ H + V , see [5]. For some positive results in the Heisenberg group, and in general Carnot groups, see however [27] and [31].…”

“…Only when γ = 1 and k = 1 the estimate follows from that in Theorem 3.2 in [9] which also holds for ε = 0. However, as we have mentioned above, the proof in [9] relies on delicate L 2 − L 2 projection estimates previously established in [32], whereas (1.8) will be derived from fairly elementary considerations exploiting some remarkable geometric properties of the operator first noted in [25], see also [33] and [31]. We mention that the reason for which we cannot take ε = 0 in (1.8) is the failure for ℓ = Q of the Hardy inequality (2.13) in Lemma 2.3.…”

Carleman estimates for a class of variable coefficient degenerate elliptic operators with applications to unique continuation

“…These results were extended to more general variable coefficient equations by Garofalo and Vassilev in [34]. One should also see the related works [28] and [32] on the Heisenberg and more general Carnot groups. We also note that a version of the monotonicity formula for B γ played an extensive role in the recent work [16] on the obstacle problem for the fractional Laplacian.…”

The polynomial growth solutions to some sub-elliptic equations on the Heisenberg group

“…Recently, in [19], we proved some Liouville-type theorems for solutions to ∆ H u = 0 on H n . Garofalo and Rotz [10] proved some Liouville-type results for sub-harmonic functions on the Carnot group.…”