#### Date of Award

8-2016

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### First Advisor

Rodrigo Banuelos

#### Committee Chair

Rodrigo Banuelos

#### Committee Member 1

Fabrice Baudoin

#### Committee Member 2

Jonathon Peterson

#### Committee Member 3

Aaron Yip

#### Abstract

Many probabilistic constructions have been created to study the *Lp*-boundedness, 1 < *p* < ∞, of singular integrals and Fourier multipliers. We will use a combination of analytic and probabilistic methods to study analytic properties of these constructions and obtain results which cannot be obtained using probability alone.

In particular, we will show that a large class of operators, including many that are obtained as the projection of martingale transforms with respect to the background radiation process of Gundy and Varapolous or with respect to space-time Brownian motion, satisfy the assumptions of Calderón-Zygmund theory and therefore boundedly map *L1* to weak-* L1.*

We will also use a method of rotations to study the *L p* boundedness, 1 < *p* < ∞, of Fourier multipliers which are obtained as the projections of martingale transforms with respect to symmetric α-stable processes, 0 < α < 2. Our proof does not use the fact that 0 < α < 2 and therefore allows us to obtain a larger class of multipliers, indexed by a parameter, 0 < *r* < ∞, which are bounded on *L p.* As in the case of the multipliers which arise as the projection of martingale transforms, these new multipliers also have potential applications to the study of the Beurling-Ahlfors transform and are related to the celebrated conjecture of T. Iwaniec concerning its exact *Lp* norm.

#### Recommended Citation

Perlmutter, Michael A., "Martingales, Singular Integrals, and Fourier Multipliers" (2016). *Open Access Dissertations*. 829.

https://docs.lib.purdue.edu/open_access_dissertations/829