The Convergence of Lebesgue Derivatives and Ergodic Averages

We study certain operators defined by infinite series that describe the nature of convergence of stochastic processes; these include square functions, oscillation operators, and variation operators. The goal is to prove that these operators map Linfinity to BMO and are of strong type (p, p) where 1 < p < infinity for the case that the stochastic processes are Lebesgue differentiation or ergodic averages. In Chapter 2, we prove the appropriate sublinear operator interpolation between the weak type (1, 1) estimate and the strong estimate from Linfinity to BMO. In Chapter 3, we prove that these operators map Linfinity to BMO and are of strong type (p, p) which 1 < p < infinity for Lebesgue derivatives. In Chapter 4, we prove that these operators for ergodic averages are of strong type (p, p) for 1 < p < infinity. In the last chapter, we characterize the strong estimate from Linfinity to L infinity for these operators. We also construct explicit counterexamples to show that the role of BMO is vital since these operators do not map Linfinity to L infinity in general.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2015

Author dc:creator

• Liu, Chaoyuan

Contributors dc:contributor

• Rosenblatt, Joseph

• Mathematics

Language dc:language

eng

Identifier

(MiAaPQ)AAI3202132

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/86857