On the Norm of the Beurling-Ahlfors Operator in Several Dimensions

http://dx.doi.org/10.4153/CMB-2010-100-0
Canad. Math. Bull. 54(2011), 113-125

  Published:2010-08-26
 Printed: Mar 2011

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 $$\|S\|_{\mathcal{L}(L^p(\mathbb{R}^n;\Lambda))}\leq(n/2+1)(p^*-1),\quad p^*:=\max\{p,p'\}$$ This improves on earlier results in all dimensions $n\geq 3$. The proof is based on the heat extension and relies at the bottom on Burkholder's sharp inequality for martingale transforms.