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

The Strong Maximum Principle and the Harnack inequality for a class of hypoelliptic non-Hörmander operators

“…Nonetheless, our results are sufficient to prime a great part of the PotentialTheory machinery needed to investigate further relevant topics concerning our PDOs L, such as Wiener-type regularity results as well as the study of the distinguished properties of the harmonic sheaf and of the subharmonic cone related to L, which will be part of future investigations, completing the results in [1,5,8,9].…”

“…We remark that topological properties similar to those mentioned above for the space of the L-harmonic functions are also valid when L in (1.1) is not necessarily hypoelliptic, provided that it possesses a global positive fundamental solution (not necessarily smooth): see e.g., [5] by the first and third named authors, where Montel-type results are proved (in the sense of [53] Acknowledgements. -We express our gratitude to Alberto Parmeggiani for many helpful discussions on hypoellipticty, leading to an improvement of the manuscript.…”

“…Another type of assumption can be made in facing potential-theoretic problems for operators L as in (1.1): indeed, very recently a systematic study of the Potential Theory for the harmonic/subharmonic functions related to L has been carried out in the series of papers [1,5,8,9], under the assumption that L possesses a smooth, global and positive fundamental solution. For the use of the fundamental solution in obtaining the Harnack Inequality for Hörmander sums of squares, see: Citti, Garofalo and Lanconelli [15]; Garofalo and Lanconelli [32,33]; Pascucci and Polidoro [58,59]; see also the recent survey by Bramanti, Brandolini, Lanconelli and Uguzzoni, [11], for the same relevant use of the fundamental solution for heat PDOs structured on Hörmander vector fields.…”

The existence of a global fundamental solution for homogeneous Hörmander operators via a global lifting method

“…5 It is sometimes customary to say that a smooth map π : M → N is a lifting of P to P if (2.4) is replaced by…”

The Role of Fundamental Solution in Potential and Regularity Theory for Subelliptic PDE

“…The use of asymptotic average operators in the characterization of classical subharmonic functions has a long history, starting with the papers [9] [21], and then by Bonfiglioli and Lanconelli, who obtained new results concerning with: Harnack-Liouville type theorems [11]; characterizations of subharmonicity [13] (see also the very recent paper [17]); average formulas and representation theorems [17]; formulas of Poisson & Jensen type; maximum principles for open unbounded sets [12]; the Dirichlet problem with L p boundary data and the Hardy spaces associated with them [14]; the Eikonal equation and Bôcher-type theorems for the removal of singularities [15]; convexity properties of the mean-value formulas with respect to the radius [18]; Gauss-Koebe and Montel type normality results [5].…”