Normal families of functions for subelliptic operators and the theorems of Montel and Koebe

A classical theorem of Montel states that a family of holomorphic functions on a domain Ω ⊆ C , uniformly bounded on the compact subsets of Ω , is a normal family. The aim of this paper is to obtain a generalization of this result in the subelliptic setting of families of solutions u to L u = 0 , where L belongs to a wide class of real divergence-form PDOs, comprising sub-Laplacians on Carnot groups, subelliptic Laplacians on arbitrary Lie groups, as well as the Laplace–Beltrami operator on Riemannian manifolds. To this end, we extend another remarkable result, due to Koebe: we characterize the solutions to L u = 0 as fixed points of suitable mean-value operators with non-trivial kernels. A suitable substitute for the Cauchy integral formula is also provided. Finally, the local-boundedness assumption is relaxed, by replacing it with L loc 1 -boundedness.