\(H\)-convex distributions in stratified groups (Q2845438)

A finite-dimensional connected and simply connected nilpotent Lie group can be thought of as a vector space \(G\), equipped with a polynomial group operation given by the Baker-Campbell-Hausdorff formula. Let \({\mathfrak g}\) be the Lie algebra of \(G\). The authors call \(G\) stratified if \({\mathfrak g}=V_1\oplus\cdots \oplus V_\iota\) with \([V_1,V_{j-1}]=V_j\) for all \(1<j\leq\iota\) and \(V_j=\{0\}\) if and only if \(j>\iota\). The authors use \(V_1\) to define certain horizontal segments in \(G\) and say that a function on an open set \(\Omega\subseteq G\) is \(h\)-convex if, for every \(x,y\in \Omega\) with \(x\) in the horizontal segment through \(y\), a certain convexity property is satisfied. On the other hand, a distributional Hessian \(D^2_HT\) is defined for each \(T\in {\mathcal D}'(\Omega)\). The distribution \(T\) is called \(h\)-convex if \(\langle D^2_HT,\psi\rangle\geq 0\) for each non-negative test function \(\psi\in {\mathcal D}(\Omega)\). The main result asserts that, if \(T\) is \(h\)-convex, then \(T\) is defined by an \(h\)-convex function on \(\Omega\).