DISCRETE AHLFORS-BEURLING TRANSFORM AND ITS PROPERTIES

Lp–BOUNDS FOR THE BEURLING–AHLFORS TRANSFORM

Let B denote the Beurling-Ahlfors transform defined on L(C), 1 < p < ∞. The celebrated conjecture of T. Iwaniec states that its L norm ‖B‖p = p∗ − 1 where p∗ = max{p, p p−1}. In this paper the new upper estimate ‖B‖p ≤ 1.575 (p − 1), 1 < p < ∞ is found.

On the Weak-type Constant of the Beurling-ahlfors Transform

(1.2) ‖B‖p = p∗ − 1, where 1 < p <∞ and p∗ = max{p, p p−1}, is partly motivated by its relation to the Gehring-Reich conjecture proved by Astala in [1]. (The lower bound of (p∗ − 1) was obtained by Lehto [16] in 1965.) Recent work has revealed B as an exemplary junction between Fourier analysis and probability, and martingale methods established by Burkholder have led to the present best known ...

The Martingale Structure of the Beurling-ahlfors Transform

The Beurling-Ahlfors operator reveals a rich structure through its representation as a martingale transform. Using elementary linear algebra and martingale inequalities, we obtain new information on this operator. In particular, Essén-type inequalities are proved for the complex Beurling-Ahlfors operator and its generalization to higher dimensions. Moreover, a new estimate of their norms is obt...

Sharp Inequalities for the Beurling-ahlfors Transform on Radial Functions

For 1 ≤ p ≤ 2, we prove sharp weak-type (p, p) estimates for the BeurlingAhlfors operator acting on the radial function subspaces of Lp(C). A similar sharp Lp result is proved for 1 < p ≤ 2. The results are derived from martingale inequalities which are of independent interest.

Martingales and the Beurling-ahlfors Operator, an Overview