In this paper, some necessary and sufficient conditions for a weighted weak type inequality of the form ∫ _{M_g^ + (f) > λ}φ (λω _1(x))ω _2(x)g(x)dx ≤C_1∫ _ - ∞^ + ∞φ (C_1 |f(x)|ω _3(x))ω _4(x)g(x)dx to hold are obtained, which generalize some known results.

In this article, John-Nirenberg inequality for Lipschitz martingale spaces is established. We further prove that Lipschitz martingale spaces Lambda(p)(alpha) are equivalent for 0 < p < infinity, which generalizes an important result in classical martingale H-p theory.

Let $$\varphi _1, \gamma $$ be nondecreasing functions on $$[0,\infty ) $$ , $$\varphi _2$$ be a quasi-convex function and $$M^+f$$ be the one-sided Hardy–Littlewood maximal function on $$\mathbb {R}^2$$ . In this paper, we give a characterization theorem for a weighted weak type inequality of the form $$\begin{aligned} \varphi _1(\lambda )\omega (\{x \in \mathbb {R}^2: M^+f(x)> \lambda \}) \le C\int _{\mathbb {R}^2}\varphi _2 \left( C\frac{|f(x)|\varrho (x)}{\gamma (\lambda )} \right) \sigma (x)dx, \end{aligned}$$ which generalizes and unifies some known results.

Let ɣ and Φ1 be nondecreasing and nonnegative functions defined on [0, ∞), and Φ2 is an N -function, u, v and w are weights. A unified version of weighted weak type inequality of the formfor martingale maximal operators f ∗ is considered, some necessary and su@cient conditions for it to hold are shown. In addition, we give a complete characterization of three-weight weak type maximal inequality of martingales. Our results generalize some known results on weighted theory of martingale maximal operators.

We collect known and prove new necessary and sufficient conditions for the weighted weak type maximal inequality of the form $${\Phi _1}(\lambda)\varrho (\{x \in X:{M_\mu }f(x) > \lambda \}) \leqslant c\int_X {{\Phi _2}(c\left| {f(x)} \right|)\sigma (x)\,{\rm{d}}\mu (x),}$$ which extends some known results.

This paper is devoted to the study of real interpolation between martingale HardyLorentz and BMO spaces in the framework of interpolation with a function parameter. We first establish some inequalities for the sharp functions of martingales. With the aid of these inequalities, some new interpolation theorems which generalize some fundamental interpolation theorems in classical martingale Hp theory are proved. In particular, we show that (Hs p0 ,q0 ,BMO2)ρ,q = Λ s q(t 1 p0 /ρ(t 1 p0 )), where 0 < p0 < ∞ , 0 < q0,q ∞ and ρ ∈ Q(0,1) . Mathematics subject classification (2010): 60G42, 60G46.

Abstract In this paper, an equivalent quasinorm for the Lipschitz space of noncommutative martingales is presented. As an application, we obtain the duality theorem between the noncommutative martingale Hardy space h p c ( ℳ ) {h}_{p}^{c}( {\mathcal M} ) (resp. h p r ( ℳ ) {h}_{p}^{r}( {\mathcal M} ) ) and the Lipschitz space λ β c ( ℳ ) {\lambda }_{\beta }^{c}( {\mathcal M} ) (resp. λ β r ( ℳ ) {\lambda }_{\beta }^{r}( {\mathcal M} ) ) for 0 < p < 1 0\lt p\lt 1 , β = 1 p − 1 \beta =\tfrac{1}{p}-1 . We also prove some equivalent quasinorms for h p c ( ℳ ) {h}_{p}^{c}( {\mathcal M} ) and h p r ( ℳ ) {h}_{p}^{r}( {\mathcal M} ) for p = 1 p=1 or 2 < p < ∞ 2\lt p\lt \infty .

In this article, some necessary and sufficient conditions are shown in order that weighted inequality of the form $$\int_{\left\{ {f* > \lambda } \right\}} {\Phi _1 \left( {\lambda w_1 } \right)w_2 d\mathbb{P}} \leqslant C\int_\Omega {\Phi _2 \left( {C\left| {f_\infty } \right|w_3 } \right)w_4 d} \mathbb{P}$$ holds a.e. for uniformly integrable martingales f = (fn)n≥0 with some constant C > 0, where Φ1, Φ2 are Young functions, wi (i = 1, 2, 3, 4) are weights, $$f* = \mathop {\sup }\limits_{n \geqslant 0} \left| {f_n } \right|$$ and $$f_\infty = \mathop {\lim }\limits_{n \to \infty } f_n$$ a.e. As an application, two-weight weak type maximal inequalities of martingales are considered, and particularly a new equivalence condition is presented.

In this paper, we shall prove a log majorization inequality, which extends Furuta’s result. c ©2017 All rights reserved.

对3类由凹函数生成的弱Orlicz鞅空间建立了相应的弱原子分解.作为应用,首先给出了这些弱Orlicz鞅空间上次线性算子有界的一个充分条件,并在此基础上证明了一些弱型鞅不等式,然后证明了关于这些弱Orlicz鞅空间的Marcinkiewicz型插值定理.

In this paper, the dual spaces of martingale Hardy Lorentz spaces H-p,H- (s)(q) are identified by use of the technique of atomic decomposition. We show that the dual spaces of martingale Hardy Lorentz spaces H-p,H- (s)(q) are A(2) (alpha), where 0 <p < 1, 0 < q <= 1, alpha = 1/p - 1.

Let f = (f(n)) (n >= 0) be a martingale, 0 < p < infinity, 1 <= q < infinity. In this paper we obtain a L-p,L-q-version of Burkholder-Gundy-Davis martingale inequalityparallel to S(f) parallel to(p,q) approximate to M(f) parallel to (p,q),by means of rearrangement technique.

In this paper, we show two Marcinkiewicz type interpolation theorems for weak Orlicz martingale spaces by employing the technique of atomic decompositions. As an application, we prove some martingale inequalities with weak Orlicz space norm.

Bounded convergence theorem is an important theorem in the theory of real variable functions.It al-ways serves as a corollary of Lebesgue dominated convergence theorem in many textbooks.We give a new proof of the bounded convergence theorem by use of the Egoroff theorem, and the conditions of the bounded convergence theorem are also discussed.

In this paper, we obtain some good λ -inequalities and rearrangement inequalities for vector-valued martingales, which are closely related to the geometric properties of the under- lying Banach space. In particular, our results extend some important inequalities in classical martingale Hp theory, and we establish a relationship between the good λ -inequality and the rearrangement inequality for some vector-valued martingale function pairs.

In this paper, some new inequalities for the sharp functions of martingales are established by use of rearrangement technique.

In this paper, we consider the real interpolation with a function parameter between martingale Hardy and BMO spaces. An interpolation theorem for martingale Hardy and BMO spaces is formulated. As an application, real interpolation between martingale Lorentz and BMO spaces is given.

We apply function parameters, which was introduced by T. F. Kalugina, to interpolation between martingale Hardy spaces. Some new interpolation theorems for Lorentz martingale spaces are proved.

In this article, we establish some atomic decomposition theorems for martingale Hardy-Lorentz spaces. As applications, with the help of weak atomic decompositions, some interpolation theorems for martingale Hardy-Lorentz spaces are proved.

In this paper,some new isomorphic characterizations of q-convex Banach spaces are given in terms of good A-inequalities and rearrangement inequalities of vector-valued martingales. In particular,we show that the inequalities are equivalent to each other when the martingales take their values in q-convexifiable Banach spaces.