The $L^p-$Operator Norm of a Quadratic Perturbation of the Real Part of the Ahlfors--Beurling Operator

TL;DR

This paper precisely calculates the $L^p$ operator norm of a quadratic perturbation of the real part of the Ahlfors--Beurling operator, introducing new methods involving laminates and heat martingales.

Contribution

It introduces a novel approach using laminates for lower bounds and extends martingale transform estimates to continuous martingales for upper bounds.

Findings

Exact $L^p$ operator norm computed.

New laminate construction for lower bounds.

Extension of martingale estimates to continuous case.

Abstract

We will explain how to compute the exact $L^p$ operator norm of a "quadratic perturbation" of the real part of the Ahlfors--Beurling operator. For the lower bound estimate we use a new approach of constructing a sequence of laminates (probability measures for which Jensen's inequality holds, but for rank one concave functions) to give an almost extremal sequence to approximate the operator. The upper bound estimate is given by extending the estimates of the quadratic perturbation of the martingale transform to continuous martingales. The use of "heat martingales" then allow us to connect the Riesz transforms to the continuous martingale estimate.