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Viscosity convex functions on Carnot groups. (English) Zbl 1057.22012
A simply connected (nilpotent) Lie group $$G$$ with Lie algebra $$\mathfrak{g}$$ decomposed as $$\mathfrak{g} = V_1 + V_2 + \cdots + V_r$$ with $$[V_1, V_j] = V_{j+1}$$ for $$j < r$$ and $$[V_1,V_r] = 0$$ is called a Carnot group. Given a domain $$\Omega\subset G$$, a function $$u:\Omega\to\mathbb{R}$$ is said to be $$h$$-convex whenever $$u| [p, q]$$ is convex for any line segment $$[p, q] = \exp ([0, 1]\cdot X)\subset\Omega$$ with $$X\in V_1$$. If $$u:\Omega\to\mathbb{R}$$ is upper semicontinuous, then $$u$$ is said to be $$v$$-convex whenever its horizontal hessian $$\nabla ^2_h u = \nabla ^2u| V_1\times V_1$$ is nonnegative in the sense of viscosity; that is $$\sum _{i,j=1}^m \xi_i\xi_jX_iX_j\phi (p)\neq 0$$ for any $$p\in\Omega$$, $$\xi_1, \dots , \xi_m\in\mathbb{R}$$ ($$m = \dim V_1$$), a basis $$X_1, \dots X_m$$ of $$V_1$$, and any C$$^2$$-function $$\phi$$ “touching” $$u$$ at $$p$$ from above. Theorem. Any upper semicontinuous $$v$$-convex function on a Carnot group is $$h$$-convex.

MSC:
 2.2e+31 Analysis on real and complex Lie groups 2.2e+26 Nilpotent and solvable Lie groups
Keywords:
Carnot group; convex function
Full Text:
References:
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