  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.

Classical Analysis and ODEs;Functional Analysis