Remarks about the Besicovitch covering property in Carnot groups of step 3 and higher (Q2790917)

The main result in this paper is Theorem~1.6, where it is shown that the Besicovitch Covering Property (BCP) does hot hold in Carnot groups of step \(3\) or higher endowed with the quasi-distances associated to the homogeneous norms with unit balls NEWLINE\[NEWLINE \{x\in G: c_1|x_1|^{\gamma_1}+\cdots +c_n|x_n|^{\gamma_n}\leq 1\}, NEWLINE\]NEWLINE or NEWLINE\[NEWLINE \{ x\in G: c_1||\bar{x}_1||_{d_1}^{\gamma_1}+\cdots +c_s||\bar{x}_s||_{d_s}^{\gamma_s}\leq 1\}. NEWLINE\]NEWLINE Here, \(x\in G\) is identified with \((x_1,\ldots,x_n)\in {\mathbb R}^n\) via exponential coordinates of the first kind on \(G\), and \(\gamma_i,c_i\) are positive constants. In the last expression, \(\bar{x}_j=(x_{m_{j-1}+1},\ldots,x_{m_j})\) is adapted to the stratification, and \(||\cdot||_{d_j}\) is the Euclidean norm in \({\mathbb R}^{d_j}\).NEWLINENEWLINEThe proof is based on two main ingredients: the validity of BCP provides a lower bound for \(\gamma_1,\ldots,\gamma_{m_1}\) in the first case, or for \(\gamma_1\) in the second one, in terms of the step of the group; then a reduction argument transfers the problem to the case of the first Heisenberg group. Results in [the authors, ``Besicovitch covering property for homogeneous distances in the Heisenberg groups'', Preprint, \url{arXiv:1406.1484}] are then applied.NEWLINENEWLINEThis is part of a series of three interesting papers by the authors on the Besicovitch Covering Property in graded groups, completed by [loc. cit.] and [the authors, ``Besicovitch covering property on graded groups and applications to measure differentiation'', Preprint, \url{arXiv:1512.04936}].