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Hessian measures on the Heisenberg group. (English) Zbl 1282.26016
In the context of the Heisenberg group, several notions of convexity have been presented by various researchers. Several authors, simultaneously, presented the notion of convexity, e.g. N. S. Trudinger and X.-J.Wang [Topol. Methods Nonlinear Anal. 10, No. 2, 225–239 (1997; Zbl 0915.35039)]. In the present paper, the authors study the properties of \(k\)-convex functions on the Heisenberg group and prove the weak continuity of the \(k\)-Hessian measure in the Euclidean space corresponding to the case that appears in [loc. cit.]. Resolving the Garofalo-Tournier conjecture, the monotonicity formula is proved in Section 3 for cases \(n>2\).

MSC:
26B25 Convexity of real functions of several variables, generalizations
28A33 Spaces of measures, convergence of measures
35J60 Nonlinear elliptic equations
35R03 PDEs on Heisenberg groups, Lie groups, Carnot groups, etc.
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