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Quasi-convex Functions in Carnot Groups*

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Abstract

In this paper, the authors introduce the concept of h-quasiconvex functions on Carnot groups G. It is shown that the notions of h-quasiconvex functions and h-convex sets are equivalent and the L ∞ estimates of first derivatives of h-quasiconvex functions are given. For a Carnot group G of step two, it is proved that h-quasiconvex functions are locally bounded from above. Furthermore, the authors obtain that h-convex functions are locally Lipschitz continuous and that an h-convex function is twice differentiable almost everywhere.

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Authors and Affiliations

  1. Department of Applied Mathematics, Hunan Institute of Science and Technology, Yueyang, 414000, Hunan, China

    Mingbao Sun

  2. Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing, 210094, China

    Mingbao Sun

  3. Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing, 210094, China

    Xiaoping Yang

Authors
  1. Mingbao Sun
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  2. Xiaoping Yang
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Correspondence to Mingbao Sun.

Additional information

*Project supported by the Science Foundation for Pure Research of Natural Sciences of the Education Department of Hunan Province (No. 2004c251), the Hunan Provincial Natural Science Foundation of China (No. 05JJ30006) and the National Natural Science Foundation of China (No. 10471063).

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Sun, M., Yang, X. Quasi-convex Functions in Carnot Groups*. Chin. Ann. Math. Ser. B 28, 235–242 (2007). https://doi.org/10.1007/s11401-005-0052-9

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  • DOI: https://doi.org/10.1007/s11401-005-0052-9

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