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https://hdl.handle.net/2142/86973
Description
Title
H(p) Spaces and Inequalities in Ergodic Theory
Author(s)
Demir, Sakin
Issue Date
1999
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
This thesis combines the theory of ergodic Hp spaces, singular integral operators and the theory of Banach space-valued operators to establish a new method to study the inequalities for the operators induced by an ergodic, measure preserving transformation. This method also indicates the close connection between the classical theory of H p spaces and ergodic Hp spaces. By means of this connection a one sided analog of a result of B. Davis for martingale square function is proven for ergodic square function, and it is shown that one can prove the same result for a large class of operators in ergodic theory by the same method. As a corollary it is shown that one can find the integrability condition for the same class of operators analog to a result of D. S. Ornstein for the ergodic maximal function. Furthermore, it is shown that one can extend the problems of classical H p spaces to the ergodic Hp spaces, and as an application of this extension a number of inequalities in ergodic theory are proven for a large class of operators. Finally, it is shown in various perspectives that one can study the vector-valued inequalities in ergodic theory as in classical harmonic analysis. To study these inequalities some methods are introduced, and by means of these methods a large class of operators in ergodic theory is discussed extensively and various types of vector-valued inequalities are proven.
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