#### 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).

#### Recommended Citation

Boros, Nicholas; Janakiraman, Prabhu; and Volberg, Alexander, "Perturbation of Burkholder's Martingale Transform and Monge-Ampère Equation" (2012). *Faculty Scholarship – Mathematics*. 4.

http://digitalcommons.olivet.edu/math_facp/4

#### Creative Commons License

This work is licensed under a Creative Commons Attribution 3.0 License.

## Comments

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