University of Illinois Urbana-Champaign

The Convergence of Lebesgue Derivatives and Ergodic Averages

Liu, Chaoyuan

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Description

Title
The Convergence of Lebesgue Derivatives and Ergodic Averages
Author(s)
Liu, Chaoyuan
Issue Date
2005
Doctoral Committee Chair(s)
Rosenblatt, Joseph
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
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.
Graduation Semester
2005
Type of Resource
text
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http://hdl.handle.net/2142/86857

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