ar X iv : 0 70 8 . 36 55 v 1 [ m at h . A P ] 2 7 A ug 2 00 7 GRADIENT REGULARITY FOR ELLIPTIC EQUATIONS IN THE HEISENBERG GROUP
نویسنده
چکیده
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 [40], 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 pLaplacean operator, extending some regularity proven in [17]. In turn, 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 [30, 14], 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.
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