Weighted integral inequalities for the maximal geometric mean operator in martingales

Under appropriate conditions on Young's functions Φ₁ and Φ₂, we give necessary and sufficient conditions in order that weighted integral inequalities hold for the maximal geometric mean operator G in martingale Orlicz classes. When Φ₁=tp and Φ₂=tp, the inequalities revert to the ones of strong or weak (p,p)-type in martingale spaces.     

Best Simultaneous Approximation in LΦ(I,X)

Let X be a Banach space and Φ be an Orlicz function.Denote by LΦ(I,X)the space of X-valued Φ-integrable functions on the unit interval I equipped with the Luxemburg norm.For f1,f2,...,fm ∈ LΦ(I,X),a distance formula disΦ(f1,f2,...,fm,LΦ(I,G))is presented,where G is a close subspace of X.Moreover,some existence and characterization results concerning the best simultaneous approximation of LΦ(I,G)in LΦ(I,X)are given.     

Necessary and sufficient conditions for weighted Orlicz class inequalities for maximal functions and singular integrals. I

Criteria of various weak and strong type weighted inequalities are established for singular integrals and maximal functions defined on homogeneous type spaces in the Orlicz classes.     

Inequalities and properties of some generalized Orlicz classes and spaces

We discuss and complement the knowledge about generalized Orlicz classes $ \tilde X_\Phi $ and Orlicz spaces X _Φ obtained by replacing the space L ¹ in the classical construction by an arbitrary Banach function space X. Our main aim is to focus on the task to study inequalities in such spaces. We prove a number of new inequalities and also natural generalizations of some classical ones (e.g., Minkowski’s, Hölder’s and Young’s inequalities). Moreover, a number of other basic facts for further study of inequalities and function spaces are included.     

鞅的(Φ_1,Φ_2)-型倒向极大不等式

设Φ_1,Φ_2是非负凸函数,证明了鞅的倒向极大算子不等式‖f‖Φ_2≤C‖f*‖Φ_1对于任意鞅f=(f_n)_n≥0成立的充分必要条件是Φ_2(?)Φ_1;鞅的极大算子均方算子的极大极小不等式‖M(f)‖Φ_2≤C_1‖m(f)‖Φ_1及‖m(f)‖Φ_2≤C_2‖M(f)‖Φ_1成立的充分必要条件是Φ_2(?)Φ_1,这里M(f)=max{f*,S(f)},m(f)=min{f*,S(f)}分别是极大算子、均方算子的极大极小函数.     

Inequalities for Hardy-type operators on the cone of decreasing functions in a weighted Orlicz space

Modular inequalities and inequalities for the norms of Hardy-type operators on the cone Ω of positive functions and on the cone of positive decreasing functions with common weight and common Young function in a weighted Orlicz space are considered. A reduction theorem for the norm of the Hardy operator on the cone Ω is obtained. It is shown that this norm is equivalent to the norm of a modified operator on the cone of all positive functions in the space under consideration. It is proved that the modified operator is a generalized Hardy-type operator. The equivalence of modular inequalities on the cone Ω and modified modular inequalities on the cone of all positive functions in the Orlicz space is shown. A criterion for the validity of such inequalities in general Orlicz spaces is obtained and refined for weighted Lebesgue spaces.     

Weighted inequalities for Hardy-type operators on the cone of decreasing functions in an Orlicz space

We establish criteria for the validity of modular inequalities for the Hardy operator on the cone Ω of nonnegative decreasing functions from weighted Orlicz spaces with general weight. The result is based on the theorem on the reduction of modular inequalities for positively homogeneous operators on the cone Ω, which enables passing to modular inequalities for modified operators on the cone of all nonnegative functions from an Orlicz space. It is shown that, for the Hardy operator, the modified operator is a generalized Hardy operator. This enables us to establish explicit criteria for the validity of modular inequalities.     

鞅的双权双&Phi|-函数极大不等式

该文研究了鞅Orlicz空间加权不等式, 主要包括弱(Φ₁,Φ₂) -型加权不等式和强(Φ₁,Φ₂) -型加权不等式. 讨论了这些不等式成立的充分必要条件.     

Modular inequalities in martingale Orlicz–Karamata spaces

In this paper, some new martingale inequalities in the framework of Orlicz‐Karamata spaces are provided. More precisely, we establish modular martingale inequalities associated with concave functions and slowly varying functions.     

Taylor's inequalities in Orlicz–Sobolev type spaces

In this paper, we obtain inequalities involving the Taylor polynomial and weak derivatives of a function in an Orlicz–Sobolev type space. Moreover, we show that any such function can be expanded in a finite Taylor series almost everywhere. As a consequence, we prove that the coefficients of any extended best polynomial $L^{\mathrm{\Phi }}$-approximation of a function on a ball almost everywhere converge to the weak derivatives of such a function when the radius tends to 0. Lastly, we get a mean convergence result of such coefficients.     

弱Orlicz鞅空间的弱原子分解及其应用

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.     

直线上函数的Banach空间的Poincaré型不等式的变分公式

基于研究对数Sobolev,Nash和其它泛函不等式的需要,将Poincare不等式 的变分公式拓广到一大类直线上函数的Banach(Orlicz)空间.给出了这些不等式成立 与否的显式判准和显式估计. 作为典型应用,仔细考察了对数Sobolev常数.     

Two Embedding Theorems in Martingale Spaces

It is proved in this paper that a pair of measurable functions satisfying the good-λ inequality yields quasi-norm inequalities in Lorentz spaces and weak Orlicz...     

Two weight inequalities for fractional one-sided maximal operators on Orlicz and Lorentz spaces

In the paper, we characterize two weight weak and strong type inequalities for the fractional one-sided maximal operators on Lorentz and Orlicz space.     

Carleson Conditions in Bergman–Orlicz Spaces

We give characterizations of (forward and reverse) Carleson conditions in terms of inequalities that involve functions in Bergman–Orlicz spaces.     

Inégalités de Faber–Krahn et Inclusion de Sobolev–Orlicz

Equivalence is demonstrated between a generalized form of the inequality of Faber–Krahn and an inequality of Sobolev–Orlicz. Equivalences with estimates on the decay of the heat kernel and inequalities on capacities and Green's functions are also given.     

Variational Formulas of Poincaré-Type Inequalities in Banach Spaces of Functions on the Line

Motivated from the study of logarithmic Sobolev, Nash and other functional inequalities, the variational formulas for Poincaré inequalities are extended to a large class of Banach (Orlicz) spaces of functions on the line. Explicit criteria for the inequalities to hold and explicit estimates for the optimal constants in the inequalities are presented. As a typical application, the logarithmic Sobolev constant is carefully examinated. Received December 13, 2001, Accepted March 26, 2002