Laminates Meet Burkholder Functions

2011/09/22 by Boros, Nicholas, Szekelyhidi, Laszlo, Volberg, Alexander · 3 citations
#Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1109.4865

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.

Cited by

• Korn's inequality from the viewpoint of calculus of variations
• Martingales, laminates and minimal Korn inequalities
• Non-uniqueness and failure of Calderón-Zygmund estimates below the critical exponent for non-monotone PDE with linear growth

Related

• The L^p-Operator Norm of a Quadratic Perturbation of the Real Part of\n the Ahlfors--Beurling Operator
• On the norm of the Beurling-Ahlfors operator in several dimensions
• The foundational inequalities of D.L. Burkholder and some of their\n ramifications
• Sharp estimates of martingale transforms in higher dimensions and applications to the Ahlfors-Beurling operator
• Stability in Burkholder's differentially subordinate martingales\n inequalities and applications to Fourier multipliers