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 all issue raised in [J.J. Manfredi. G. Mingione, Regularity results tor quasilinear elliptic equations in the Heisenberg group, 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, C-L alpha-regularity for p-harmonic functions in the Heisenberg group for p near 2, in: Contemp. Math.. vol. 370, 2005, pp. 17-23]. In turn, using some recent techniques of Caffarelli and Peral [L. Caffarelli. I. Peral, On W-Lp estimates for elliptic equations in divergence Corm, Comm. Pure Appl. Math. 51 (1998) 1-21], the a priori estimates found are shown to imply the Suitable local Calderon-Zygmund theory for the related class of non-homogeneous, possibly degenerate equations, involving discontinuous coefficients. These last results extend to the sub-elliptic setting a few classical non-linear Euclidean results [T. Iwaniec. Projections onto gradient fields and L-p-estimates for degenerated elliptic operators, Studia Math. 75 (1983) 293-312; E. DiBenedetto, J.J. Manfredi, Oil the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems. 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. (C) 2009 Elsevier Inc. All rights reserved.
