New estimates for the beurling-ahlfors operator on differential forms

We establish new Lp estimates for the norm of the generalized Beurling-Ahlfors transform S acting on form-valued functions. Namely, we prove that {double pipe}S{double pipe}_{Lp(R{double-struck}n;Λ)} ≤ n(p^*-1) where p^*= max{p, p/(p -1)}, thus extending the recent Nazarov-Volberg estimates to higher dimensions. The even-dimensional case has important implications for quasiconformal mappings. Some promising prospects for further improvement are discussed at the end.