Laminates Meet Burkholder Functions

TL;DR

This paper computes the exact $L^p$ operator norm of a quadratic perturbation of the Ahlfors--Beurling operator using laminates for lower bounds and heat martingales for upper bounds, advancing understanding of singular integral operators.

Contribution

It introduces a novel laminate construction for lower bounds and extends martingale transform estimates to continuous martingales for upper bounds in operator norm calculation.

Findings

Exact $L^p$ norm of the quadratic perturbation is determined.

Laminates effectively approximate extremal sequences for lower bounds.

Heat martingales connect Riesz transforms to continuous martingale estimates.

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.