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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.

22E30 Analysis on real and complex Lie groups
22E25 Nilpotent and solvable Lie groups
Full Text: DOI arXiv
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