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Quasi-convex functions in Carnot groups. (English) Zbl 1123.43004
The authors introduce the concept of \(h\)-quasiconvexity which generalizes the notion of \(h\)-convexity in the Carnot group \(G\). An example of \(h\)-quasiconvex function which is not \(h\)-convex is provided. Some interesting properties similar to those of \(h\)-convex functions on \(G\) are given. In particular, the authors show that the notions of \(h\)-quasiconvex functions and \(h\)-convex sets are equivalent, give the \(L^\infty\) estimates of the first derivatives of \(h\)-quasiconvex functions, and prove that \(h\)-quasiconvex functions on a Carnot group \(G\) of step two are locally bounded from above. Furthermore, for a Carnot group \(G\) of step two, the authors obtain that \(h\)-convex functions are locally Lipschitz continuous and twice differentiable almost everywhere in \(G\). This last result is the version of the Busemann-Feller-Alexandrov theorem for the class of \(h\)-convex functions in Carnot groups of step two.

43A80 Analysis on other specific Lie groups
43-06 Proceedings, conferences, collections, etc. pertaining to abstract harmonic analysis
26B25 Convexity of real functions of several variables, generalizations
Full Text: DOI
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