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# Communications in Analysis and Geometry

## Volume 11 (2003)

### Number 2

### Notions of Convexity in Carnot Groups

Pages: 263 – 341

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

#### Authors

#### 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*_{i}*X*_{j}*u + X*_{j}*X*_{i}*u} ≥ F*

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

* sup*_{ Ω}* u ≤ sup*_{ ∂Ω} *u>*^{+} *+ C *∥F∥_{LQ(Ω)} .

Here, *L*^{2,Q}_{loc} (Ω) indicates the Sobolev space of functions u ∈ *L*^{Q} _{loc}(Ω) having weak derivatives *X*_{i}*X*_{j}*u* ∈ *L*^{Q} _{loc }(Ω). We note explicitly that

*Lu = tr {A [X*^{2}*u]*^{*}},

where we have denoted by *[X*^{2}*u]*^{*} 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 =* ℝ

^{n}with the standard homogeneous dilations, one has

**g**=**V**_{1}= ℝ

^{n}, so that

*Q = n*, and

*[X*

^{2}

*u]*

^{*}=

*D*

^{2}

*u*, 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.