Weighted Inequalities In Orlicz Martingale Classes

The theory of Ap was spurred in the 1970s, from which a better understand-ing of weighted inequalities was obtained. Subsequently, the relevant subjects were quickly established in Rn, for example, factorization of Ap weights, vector-valued inequalities, extrapolation of operators, and so on. Recently, weighted inequalities were widespreadly studied in Orlicz function classes, Lorentz spaces, rearrangement invariant spaces, as well as Musielak-Orlicz function classes. In martingale spaces, weighted inequalities first appeared in 1970s, but they has been developing slowly. One reason is that some decomposition theorems and covering theorems which depend on algebraic structure and topological structure are invalid on probability space.In this paper, weighted inequalities and characterizations of weights are obtained in martingale spaces. The maximal operator, the general maximal operator and the maximal geometric mean operator are extensively studied.For the maximal operator, we construct different types of weighted inequali-ties relying on the property ofΦfunction in Orlicz martingale classes. Sometimes, we are interested in some weak inequality as well as extra-weak modular inequal-ity and at other times we are interested in integral inequalities. Specially, in some extreme case, the weak modular inequality is valid if and only if the weights are A1 ones, which coincides the well known result that the maximal operator is bounded from L1(u) to wL1(v) if and only if (u,v)∈A1. Moreover, for a special class of two-parameter martingale spaces, exploiting extrapolation theory as a tool, we obtain a mixed-norm weighted inequality.We introduce the general maximal operator M into martingale space, and character the weighted inequality, too. In fact, if p>1, then M maps Lp(Ω) to Lp(Ω×N,μ) or wLp(Ω×N,μ) if and only ifμis a Carleson measure onΩ×N. In martingale setting, the theory of Carleson measure is further enriched by us.Compared with the maximal operator, the maximal geometric mean oper-ator is neither a sub-linear operator nor a quasi-linear operator. Therefore, the interpolation theory is invalid for it. Considering its feature, we obtain some weighted integral inequalities in Orlicz martingale classes. WhenΦis a power function, our inequalities reduce to the (p, q) ones, which are very interesting. In fact, the weighted inequalities depend only on the q/p rather than depend solely on indicators of p and q; on the other hand, the operator maps L1 to L1. While studying the maximal geometric mean operator, we also focus our attention on the A∞weight. On the assumption thatω∈S, we give a variety of equivalent definitions ofω∈A∞.The paper is divided into five chapters. The first one surveys results of mar-tingale spaces and weights. It also contains the significance of the dissertation. Chapter 2 consists of preliminaries. The following Chapter 3 discusses equiva-lent definitions of A∞weight. In addition, a class of two-parameter martingale spaces is studied. Chapter 4 is devoted to weighted inequalities for the maximal operator in Orlicz martingale classes. Moreover, the Carleson measure and the general maximal operator are also considered. The last chapter deals with the maximal geometric mean operator.