*Published Paper*

**Inserted:** 29 sep 2012

**Last Updated:** 29 sep 2012

**Journal:** Adv. Math.

**Volume:** 222

**Pages:** 62-129

**Year:** 2009

**Abstract:**

We give dimension-free regularity conditions for a class of possibly degenerate sub-elliptic equations in the Heisenberg group exhibiting super-quadratic growth in the horizontal gradient; this solves an issue raised in J.J. Manfredi, G. Mingione (Math. Ann. 339 (2007) 485–544), where only dimension dependent bounds for the growth exponent are given. We also obtain explicit a priori local regularity estimates, and cover the case of the horizontal p-Laplacean operator, extending some regularity proven in A. Domokos, J.J. Manfredi, (Contemp. Math., vol. 370, 2005, pp. 17–23). In turn, using some recent techniques of Caffarelli and Peral (Comm. Pure Appl.Math. 51 (1998) 1–21), the a priori estimates found are shown to imply the suitable local Calderón–Zygmund theory for the related class of non-homogeneous, possibly degenerate equations involving discontinuous coefficients. These last results extend to the subelliptic setting a few classical non-linear Euclidean results T. Iwaniec (Studia Math. 75 (1983) 293–312 and E. DiBenedetto, J.J. Manfredi (Amer. J. Math. 115 (1993) 1107–1134), and to the non-linear case estimates of the same nature that were available in the sub-elliptic setting only for solutions to linear equations.