Perturbation of Burkholder's Martingale Transform and Monge-Ampère Equation

Authors

Nicholas Boros, Olivet Nazarene UniversityFollow
Prabhu Janakiraman, Michigan State UniversityFollow
Alexander Volberg, Michigan State UniversityFollow

Document Type

Article

Peer Reviewed

1

Publication Date

2012

Scholarship Domain(s)

Scholarship of Discovery

Abstract

Given a sequence of martingale differences, Burkholder found the

sharp constant for the Lp-norm of the corresponding martingale transform. We

are able to determine the sharp Lp-norm of a small "quadratic perturbations"

of the martingale transform in Lp. By "quadratic perturbation" of the martin-

gale transform we mean the Lp norm of the square root of the squares of the

martingale transform and the original martingale (with small constant). The

problem of perturbation of martingale transform appears naturally if one wants

to estimate the linear combination of Riesz transforms (as, for example, in the

case of Ahlfors{Beurling operator).

Comments

Advances in Mathematics 230(s 4–6):2198–2234. DOI:10.1016/j.aim.2012.03.035

Recommended Citation

Boros, Nicholas; Janakiraman, Prabhu; and Volberg, Alexander, "Perturbation of Burkholder's Martingale Transform and Monge-Ampère Equation" (2012). Faculty Scholarship – Mathematics. 4.
https://digitalcommons.olivet.edu/math_facp/4