Sharp Logarithmic Bounds for Beurling-Ahlfors Operator Restricted to the Class of Radial Functions

Abstract

We establish a class of sharp logarithmic estimates for the Beurling-Ahlfors transform B on the complex plane. For any K > 0 we determine the optimal constant $${L = L(K) \in (0, \infty]}$$ such that the following holds. If $${F : \mathbb{C} \rightarrow \mathbb{C}}$$ is a radial function, then for any R > 0, $$\frac{1}{|\mathcal{B}(0, R)|} \int_{\mathcal{B}(0, R)} |BF(z)| dz \leq \frac{K}{|\mathcal{B}(0, R)|} \int_{\mathcal{B}(0, R)} \Psi(|F(z)|) dz + L(K),$$ where Ψ(t) = (t + 1) log(t + 1) – t and $${\mathcal{B}(0, R) \subset \mathbb{C}}$$ denotes the ball of center 0 and radius R. A related result in higher dimensions is also established. The proof rests on probabilistic methods and exploits a certain sharp inequality for martingales.