Sharp estimate of the Ahlfors-Beurling operator via averaging martingale transforms

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

Bellman function, Littlewood-Paley estimates and asymptotics for the Ahlfors-Beurling operator in L(C)

Estimation of L norms of Fourier multipliers is known to be hard. It is usually connected to some interesting types of PDE, see several such PDE for several Fourier multipliers on the line in a recent paper of Kalton and Verbitsky [13]. Sometimes, but much more rarely, one can establish sharp L estimates for Fourier multipliers in several variables. Riesz transforms are examples of success. The...

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.