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

TL;DR

This paper establishes an improved upper bound for the Lp operator norm of the generalized Beurling-Ahlfors transformation in multiple dimensions, utilizing heat extension and martingale inequalities.

Contribution

It provides a tighter bound for the operator norm in higher dimensions, advancing previous results and employing novel analytical techniques.

Findings

Operator norm bound is at most (n/2+1)(p-1) for p>2

Improves upon earlier bounds for all dimensions n>2

Uses heat extension and Burkholder's inequality in the proof

Abstract

The Lp operator norm of the generalized Beurling-Ahlfors transformation in n variables is at most (n/2+1)(p-1) for p>2. This improves on earlier results in all dimensions n>2. The proof is based on the heat extension and relies at the bottom on Burkholder's sharp inequality for martingale transforms.