Analysis of the Ahlfors-Beurling operator (lecture notes for the summer school at the University of Seville, 2013)

Two weight commutators for Beurling--Ahlfors operator

We establish the equivalent characterisation of the weighted BMO space on the complex plane $\mathbb{C}$ via the two weight commutator of the Beurling--Ahlfors operator with a BMO function. Our method of proofs relies on the explicit kernel of the Beurling--Ahlfors operator and the properties of Muckenhoupt weight class.

The L^p-norms of the Beurling-Ahlfors transform on radial functions

We calculate the norms of the operators connected to the action of the Beurling-Ahlfors transform on radial function subspaces introduced by Bañuelos and Janakiraman. In particular, we find the norm of the Beurling-Ahlfors transform acting on radial functions for $p>2$, extending the results obtained by Bañuelos and Janakiraman, Bañuelos and Osȩkowski, and Volberg for $1<p\leq 2$.

关于多维Beurling-Ahlfors算子的范数

On the norm of the Beurling-Ahlfors operator in several dimensions

当p>2时，n元广义Beurling-Ahlfors变换的Lp算子范数至多为（n/2+1）（p-1）.这改进了所有维度n>2的早期结果。证明是基于热扩展，并依赖于底部的伯克霍尔德的尖锐不等式的鞅变换。

Ahlfors-Beurling operator on radial functions

We give a simple proof of a sharp bound of Ahlfors--Beurling operator on complex-valued radial functions. In the language of the Calculus of Variation we prove a certain estimates for stretch and twist functions. Our estimate go slightly beyond this case. This article was written in February 2009 and then delivered at a seminar talk at UW, Madison on February, 2009.

关于differential forms上Beurling-Ahlfors算子的新估计

New estimates for the Beurling-Ahlfors operator on differential forms

本文建立了作用于形式值函数的广义Beurling-Ahlfors变换$\mathcal{S}$的范数的新的$p$ -估计。也就是说，我们证明，$\norm{\mathcal{S}}_{L^p(\R^n;Λ)\to L^p(\R^n;Λ)}\leq n(p^{*}-1)$，其中$p^*=\max\{p, p/(p-1)\},$，从而延长最近Nazarov-Volberg估计更高的维度。偶维情形对拟共形映射有重要的意义。最后讨论了进一步改进的前景。

Sharp estimates for the commutators of the Hilbert, Riesz transforms and the Beurling-Ahlfors operator on weighted Lebesgue spaces

We prove that the operator norm on weighted Lebesgue space L2(w) of the commutators of the Hilbert, Riesz and Beurling transforms with a BMO function b depends quadratically on the A2-characteristic of the weight, as opposed to the linear dependence known to hold for the operators themselves. It is known that the operator norms of these commutators can be controlled by the norm of the commutator with appropriate Haar shift operators, and we prove the estimate for these commutators. For the shift operator corresponding to the Hilbert transform we use Bellman function methods, however there is n…