H(p) Spaces and Inequalities in Ergodic Theory

106 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1999.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.U of I OnlyRestricted to the U of I community idenfinitely during batch ingest of legacy ETD