Communications in Analysis and Geometry

Volume 11 (2003)

Number 2

Notions of Convexity in Carnot Groups

Pages: 263 – 341

DOI: https://dx.doi.org/10.4310/CAG.2003.v11.n2.a5

Authors

Donatella Danielli

Nicola Garofalo

Duy-Minh Nhieu

Abstract

Conjecture: Given a connected, bounded open set Ω ⊂ G, let F ∈ L^Q (Ω). Suppose that u ∈ L^2,Q_loc(Ω) ∩ C(\bar Ω) satisfy

Lu ≝ 1/2 ∑ ^m _i,j=1 a_ij{ X_iX_ju + X_jX_iu} ≥ F

in Ω. There exists a constant C = C(G, ν, Ω) \gt 0 such that

sup _Ω u ≤ sup _∂Ω u\gt ⁺ + C ∥F∥_L^Q(Ω) .

Here, L^2,Q_loc (Ω) indicates the Sobolev space of functions u ∈ L^Q _loc(Ω) having weak derivatives X_iX_ju ∈ L^Q _loc (Ω). We note explicitly that

Lu = tr {A [X²u]^*},

where we have denoted by [X²u]^* the symmetrized horizontal Hessian of u, see (1.3), or Section 5. Concerning the optimality of the L^Q norm in the estimate (1.1), we refer the reader to [DGN4]. In the abelian case, when G = ℝⁿ with the standard homogeneous dilations, one has g = V₁ = ℝⁿ, so that Q = n, and [X²u]^* = D²u, the standard Hessian matrix of u. The above conjecture, in this situation, is in fact the celebrated geometric maximum principle of Alexandrov-Bakelman-Pucci (except that the matrix (a_ij) need not be uniformly elliptic), see [A], [Ba1], [Pu], and also [Ba3].

Given the pervasive role that convexity plays in such principle, in order to advance our comprehension of the above conjecture it seems natural to examine appropriate notions of convexity in the non-abelian setting. This is the goal of the present paper.

Published 1 January 2003