Regularity estimates for convex functions in Carnot-Carathéodory spaces (Q346650)

At first the notions of \(\chi\)-convex and \(v\)-convex functions with respect to a set of \(C^2\) smooth vector fields \(\chi\) are introduced. Then it is assumed that \(\chi\) satisfies the Hörmander commutator condition and the metric balls \(B_{x,r}\) are defined via the Carnot-Carathéodory distance \(d\). To be precise we formulate the main result of the paper.NEWLINENEWLINE NEWLINETheorem 1.1. Let \(K\Subset\Omega\subset\mathbb{R}^n\). Then there exist \(C(K)>0\) and \(R(K)>0\) such that each \(\chi\)-convex function \(u\) that is locally bounded from above for every \(x\in K\) satisfies the estimate: NEWLINE\[NEWLINE\sup_{B_{x,r}}|u|\leq C -\mkern-20mu\int_{B_{x,2r}}|u(w)|\,dw,\quad 0<r< RNEWLINE\]NEWLINE and a similar estimate holds for \(|u(y)- u(z)|\) for \(y,z\in B_{x,r}\). NEWLINENEWLINENEWLINEIn a corresponding second estimate, the factor \({d(y,z)\over r}\) appears. Moreover, the authors show that for every \(x\in\Omega\) the \(\chi\)-convex function \(u\) is subharmonic with respect to a suitable sub-Laplacian that is constructed around \(x\).