Maximum and Comparison Principles for Convex Functions on the Heisenberg Group

The purpose in this paper is to establish pointwise estimates for a class of convex functions on the Heisenberg group. An integral estimate for classical convex functions in terms of the Monge–Ampère operator det D2u was proved by Aleksandrov, see [3, Theorem 1.4.2]. Such estimate is of great importance in the theory of weak solutions for the Monge–Ampère equation, and its proof revolves around the geometric features of the notion of normal mapping or subdifferential in Rn [3, Definition 1.1.1] which yield in addition the useful comparison principle for Monge-Ampère measures, [3, Theorem 1.4.6]. On the Heisenberg group, and more generally in Carnot groups, several notions of convexity have been introduced and compared in [2] and [4]. The notion of convex function we use in this paper is given in Definition 2.2, and a natural question is if similar comparison and maximum principles hold in this setting. A reason for this question is that those estimates would be useful in the study of solutions for nondivergence equations of the form ai jXi X j where ai j is a uniformly elliptic measurable matrix and Xi are the Heisenberg vector fields. The difficulty for this study is the doubtful existence of a notion of normal mapping in Hn suitable to establish maximum and comparison principles. In this paper we address this question and follow a route different from the one described above for convex functions, and in particular, we do not use any notion of normal mapping. This approach was recently used by Trudinger and Wang to study Hessian equations [6]. Our integral estimates are in terms of the following Monge–Ampère type