Published online by Cambridge University Press: 20 November 2018
The generalized Beurling–Ahlfors operator $S$ on ${{L}^{p}}({{\mathbb{R}}^{n}};\,\Lambda )$, where $\Lambda \,:=\,\Lambda ({{\mathbb{R}}^{n}})$ is the exterior algebra with its natural Hilbert space norm, satisfies the estimate
This improves on earlier results in all dimensions $n\,\ge \,3$. The proof is based on the heat extension and relies at the bottom on Burkholder's sharp inequality for martingale transforms.