Generalized H (?) Lder Inequality And Its Application

Weighted inequalities arised naturally in Fourier analysis, and it had attracted much attention due to its closely relationship with a variety of subjects. The theory of Ap was started in the 1970s, from which we had a new understanding of weighted inequalities. Then the relevant results were quickly established in R", for example, vector-valued inequalities, extrapolation of operators. The multilinear weighted inequalities in harmonic analysis developed rapidly in recent years. In martingale spaces, weighted inequalities first appeared in 1970s, but they have been developing slowly. One reason was that some decomposition theorems and covering theorems which depend on algebraic structure and topological structure were invalid on probability space. So we need a new way to study the weighted inequalities in martingale spaces.In this paper, the weighted inequalities involving infinite product are obtained in martingale spaces. Firstly, we give the generalized H(?)lder’s inequality for integral. This inequality involves the infinite product, so we need to investigate the convergence of the infinite product. Proving the inequality, we do not assume the measure is a-finite. Secondly, we give the generalized H(?)lder’s inequality for conditional expectation. This proof depends on the generalized H(?)lder’s inequality for integral. Finally, using the generalized inequalities mentioned above, we establish the weighted inequalities involving infinite product. Specifically, we give the strong-type and weak-type weighted inequalities for the generalized Doob’s maximal operator, respectively. Because probability space doesn’t have algebraic structure and topological structure, the theory is more general in martingale spaces.This paper contains three chapters. The first chapter consists of the introduction and preliminary. We introduce the weighted inequalities for the Hardy-Littlewood maximal operator and the multisubliear maximal operator in R". In chapter 2, we devote to prove the generalized H(?)lder’s inequalities for integral and conditional expectation in details, which will be used in chapter 3. In chapter 3, we assume that the weight satisfies the condition of the reverse H(?)lder’s inequality, and give the weighted inequalities involving infinite product.