| In this thesis we first implement iteration methods for fractional difference quotients of weak solutions to the p-Laplace equation in the Heisenberg group. We obtain that Tu ∈ Lploc (O) for 1 < p < 4, where u is a p-harmonic function. Then we give detailed proofs for HW 2,2-regularity for p in the range 2 ≤ p < 4 and HW2, p-regularity in the case 17-12 ≤ p ≤ 2 for epsilon-approximate p-harmonic functions in the Heisenberg group. These last estimates however are not uniform in epsilon. The method to prove uniform estimates is based on Cordes type estimates for subelliptic linear partial differential operators in non-divergence form with measurable coefficients in the Heisenberg group. In this way we establish interior HW2,2-regularity for p-harmonic functions in the Heisenberg group Hn for p in an interval containing 2. We will also show that the C1,alpha regularity is true for p in a neighborhood of 2.; In the last chapter we extend our results to the more general case of Carnot groups. |
