Gradient regularity for elliptic equations in the Heisenberg group

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, Regularity results for 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 1 , α -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 1 , p estimates for elliptic equations in divergence form, 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 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, On 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.