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Regularity of convex functions on Heisenberg groups. (English) Zbl 1121.43007
The Heisenberg nilpotent Lie groups arise by releasing the quantum–mechanically motivated Heisenberg commutation relations from their operator realizations in Hilbert space [H. Weyl, The Theory of Groups and Quantum Mechanics. Dover, New York (1949; Zbl 0041.56804), A. Borel, Hermann Weyl and Lie groups. Hermann Weyl Centenary Lectures, Berlin: Springer-Verlag (1986; Zbl 0599.01014), pp. 53–82; Collected Papers, Vol. IV, New York: Springer-Verlag (2001; Zbl 1065.01016)]. Apart from their importance for the foundations of string theory [J. Lepowski, Perspectives on vertex operators and the monster. The Mathematical Heritage of Hermann Weyl, Proc. Symp. Pure Math. 48, 181–197 (1988; Zbl 0662.17007), W. Schempp [Indistinguishability and quantum entanglement, to be published in Int. J. Pure Appl. Math.], the unitary representation theory of Heisenberg Lie groups forms a highly interesting topic of mathematical research [A. Weil, Acta Math. 111, 143–211 (1964; Zbl 0203.03305); Collected Papers, Vol. III, New York: Springer-Verlag (1979; Zbl 0424.01029), pp. 1–69, A. Borel, N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups. Annals of Mathematics Studies, 94. Princeton, New Jersey: Princeton University Press (1980; Zbl 0443.22010)].
In the vein of analysis and sub–Riemannian geometry of Heisenberg groups [A. Bellaïche, Prog. Math. 144, 1–78 (1996; Zbl 0862.53031), W. Schempp, Math. Methods Appl. Sci. 22, No. 11, 867–922 (1999; Zbl 0964.92026)], the paper under review is concerned with differentiability properties of convex functions [P. Pansu, Ann. Math. (2) 129, No. 1, 1–60 (1989; Zbl 0678.53042)]. It establishes that the concepts of horizontal convexity and viscosity convexity are equivalent notions [D. Danielli, N. Carofalo and D.-M. Nhieu, Commun. Anal. Geom. 11, No. 2, 263–341 (2003; Zbl 1077.22007)], and that horizontally convex functions are locally Lipschitz continuous with respect to the Carnot-Carathéodory metric [M. Gromov, Prog. Math. 144, 79–323 (1996; Zbl 0864.53025)]. In addition, the article exhibits horizontally convex functions of Weierstraß type which are nowhere differentiable on a dense set of vertical lines.

43A80 Analysis on other specific Lie groups
26B25 Convexity of real functions of several variables, generalizations
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
Full Text: EuDML
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