New estimates for the Beurling-Ahlfors operator on differential forms

We establish new $p$-estimates for the norm of the generalized Beurling–Ahlfors transform $\mathcal{S}$ acting on form-valued functions. Namely, we prove that $\norm{$\mathcal{S}$}_{L^p(\R^n;\Lambda)\to L^p(\R^n;\Lambda)}\leq n(p^{}-1)$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.