The geometric Sobolev embedding for vector fields and the isoperimetric inequality

Smooth approximation for intrinsic Lipschitz functions in the Heisenberg group

“…which implies (10). To prove (11), we use the fact, recalled in in Definition 3.4, that the cone opening is the inverse of the Lipschitz constant, and the estimate of the cone opening provided in Theorem 3.5, with k = −…”

Basic properties of nonsmooth Hörmander's vector fields and Poincaré's inequality

“…This means that, for instance, one assumes axiomatically the validity of a connectivity theorem, a doubling property for the metric balls, a Poincaré's inequality for the "gradient" defined by the system of vector fields, and proves as a consequence other interesting properties of the metric or of second order PDE's structured on the vector fields. A good deal of papers have been written in this spirit; we just quote some of the Authors and some of the papers on this subject, which are a good starting point for further bibliographic references: Capogna, Danielli, Franchi, Gallot, Garofalo, Gutierrez, Lanconelli, Morbidelli, Nhieu, Serapioni, Serra Cassano, Wheeden; see [1], [9], [14], [15], [22], [23], [24], [33]; see also the already quoted paper [52] and the one by Hajlasz-Koskela [25].…”

Subelliptic p -harmonic maps into spheres and the ghost of Hardy spaces

“…The imbedding for compactly supported functions is easier and has been obtained in Rotschild and Stein [62], Capogna, Danielli, and Garofalo [5], [6], Danielli [16], and Franchi, Gallot, and Wheeden [24]. Note that Sobolev inequality (11) implies the Poincaré inequality…”