Weighted Inequalities Involving Infinite Products

Theories of the Ap weight and Sp weight were given in Euclidean space in the 1970s and 1980s. Since then weighted theory has been being studied. As is well known, dyadic techniques are fundamental in weighted theory. On one hand, dyadic decomposition of Rn is a powerful stopping-time construction, which has many interesting applications. On the other hand, the dyadic techniques are links between weighted inequalities in Harmonic analysis and weighted inequalities in martingale spaces, then some ideas in harmonic analysis become transparent when viewed from probabilistic angle.Recently, the multilinear weighted theory was widespreadly studied and dyadic techniques played an important role in the theory. The multilinear version of Ap theorem was given, which is sharp in some sense. But, it is difficult to establish the generalization of Sawyer’s theorem to the multilinear setting. In order to solve the problem, a kind of monotone property and a reverse Holder’s inequality on the weights were introduced. Under one of the assumptions made above, multilinear version of Sp theorem is valid.Inspired by the multi-linear weighted theory, our paper studies weighted inequalities involving infinite product. We define a generalized dyadic maximal operator and study the weighted inequalities for it. Specifically, we give the relevant Carleson embedding theorem and the weak type of generalized Holder’s inequality. Then, we character the strong type and weak type of weighted inequalities, respectively. Our results mainly depend on generalized Holder’s inequality for integral and the weak type of generalized Holder’s inequality. Because our theorems involve the infinite product, we must pay attention to the convergence of the infinite product.The paper is divided into three chapters. The first one surveys weighted theory of Hardy-Littlewood maximal operator and multilinear maximal operator in Rn. The second one consists of preliminaries, which contains some basic definitions and facts. In addition, we give the weak type of generalized Holder’s inequality in the chapter. The last chapter is devoted to our main results, which are weighted inequalities involving the infinite product.