On Hardy - Littlewood Maximal Functions in Orlicz Spaces

Let Φ(t) and Ψ(t) be the functions having the following representations Φ(t) = ∫a(s)ds and Ψ(t) = ∫b(s) ds, where a(s) is a positive continuous function such that ∫a(s)/s ds = + ∞ and b(s) is an increasing function such that lim_s→ ∞ b(s) = + ∞. Then the following statements for the Hardy - Littlewood maximal function M f (x) are equivalent:

• 1 (i) there exist positive constants c₁ and s₀ such that
• 1 (ii) there exist positive constant c₂ and c₃ such that

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Weighted Inequalities for Certain Maximal Functions in Orlicz Spaces

Let M_g be the maximal operator defined by $$M_g f\left( x \right) = \sup \frac{{\int_a^b {f\left( y \right)g\left( y \right){\text{d}}y} }}{{\int_a^b {g\left( y \right){\text{d}}y} }}$$ , where g is a positive locally integrable function on R and the supremum is taken over all intervals [a,b] such that 0≤a≤x≤b/η(b?a), here η is a non-increasing function such that η (0) = 1 and $\mathop {{\text{lim}}}\limits_{t \to {\text{ + }}\infty } \eta \left( t \right) = 0$ η (t) = 0. This maximal function was introduced by H. Aimar and L. L. Forzani [AF]. Let Φ be an N - function such that Φ and its complementary N - function satisfy Δ₂. It gives an A′_Φ(g) type characterization for the pairs of weights (u,v) such that the weak type inequality $$u\left( {\left\{ {x \in {\text{R}}\left| {M_g f\left( x \right) >\lambda } \right.} \right\}} \right) \leqslant \frac{C}{{\Phi \left( \lambda \right)}}\int_{\text{R}} {\Phi \left( {\left| f \right|v} \right)} $$ holds for every f in the Orlicz space L_Φ(v). And, there are no (nontrivial) weights w for which (w,w) satisfies the condition A′_Φ(g).     

Weighted Inequalities for Certain Maximal Functions in Orlicz Spaces

Let M_g be the maximal operator defined by

Commutators of Some Maximal Functions with Lipschitz Function on Orlicz Spaces

Let \(0\le \alpha <n\), \([b,M_{\alpha }]\) and \([b,M^{\sharp }]\) be the nonlinear commutators of the fractional maximal function \(M_{\alpha }\) and the sharp maximal function \(M^{\sharp }\) with a locally integrable function b. In this note, we give necessary and sufficient conditions for the boundedness of \([b,M_{\alpha }]\) and \([b,M^{\sharp }]\) on Orlicz spaces when the symbol b belongs to Lipschitz spaces, by which some new characterizations of non-negative Lipschitz functions (in terms of the Orlicz norm) are obtained.     

Orlicz空间中迭代极大函数的一个反加权不等式

研究了Orlicz空间中迭代Hardy-Littlewood极大函数的反加权不等式的一个等价条件,利用权函数以及分布函数的性质,结果推广了已有的关于Hardy-Littlewood极大函数的反加权不等式的结论.     

Commutators of Fractional Maximal Operator on Orlicz Spaces

In the present paper, we give necessary and sufficient conditions for the boundedness of commutators of fractional maximal operator on Orlicz spaces. The main advance in comparison with the existing results is that we manage to obtain conditions for the boundedness not in integral terms but in less restrictive terms of supremal operators.     

On Interpolation of the Fourier Maximal Operator in Orlicz Spaces

Let and be positive increasing convex functions defined on [0, ). Suppose satisfies the 2-condition, that is, (t)2 (C1t) for sufficiently large t, and has some nice properties. If -1(u)log(u+1) C2-1(u) for sufficiently large uthen we have S*(f) L CfL for all f L ([-, ])where S*(f) is the majorant function of partial sums of trigonometric Fourier series and fL is the Orlicz norm of f. This result is sharp.     

A Notion of Orlicz Spaces for Vector Valued Functions

The notion of the Orlicz space is generalized to spaces of Banach-space valued functions. A well-known generalization is based on N-functions of a real variable. We consider a more general setting based on spaces generated by convex functions defined on a Banach space. We investigate structural properties of these spaces, such as the role of the delta-growth conditions, separability, the closure of , and representations of the dual space.     

Boundedness Characterization of Maximal Commutators on Orlicz Spaces in the Dunkl Setting

On the real line, the Dunkl operators$$D_{\nu}(f)(x):=\frac{d f(x)}{dx} + (2\nu+1) \frac{f(x) - f(-x)}{2x}, ~~ \quad\forall \, x \in \mathbb{R}, ~ \forall \, \nu \ge -\tfrac{1}{2}$$are differential-difference operators associated with the reflection group $\mathbb{Z}_2$ on $\mathbb{R}$, and on the $\mathbb{R}^d$ the Dunkl operators $\big\{D_{k,j}\big\}_{j=1}^{d}$ are the differential-difference operators associated with the reflection group $\mathbb{Z}_2^d$ on $\mathbb{R}^{d}$.In this paper, in the setting $\mathbb{R}$ we show that $b \in BMO(\mathbb{R},dm_{\nu})$ if and only if the maximal commutator $M_{b,\nu}$ is bounded on Orlicz spaces $L_{\Phi}(\mathbb{R},dm_{\nu})$. Also in the setting $\mathbb{R}^{d}$ we show that $b \in BMO(\mathbb{R}^{d},h_{k}^{2}(x) dx)$ if and only if the maximal commutator $M_{b,k}$ is bounded on Orlicz spaces $L_{\Phi}(\mathbb{R}^{d},h_{k}^{2}(x) dx)$.     

Orlicz空间的小波

利用小波系数给出Orlicz空间的一个特征,并研究小波展开的无条件收敛性.     

Orlicz空间的P凸性

赋Orlicz范数的Orlicz空间的P凸等价于自反。     

Integrability Properties of the Maximal Operator on Partial Sums of Fourier Series in Orlicz Spaces

Let S* (f be the majorant function of the partial sums of the trigonometric Fourier series of f. In this paper we consider the Orlicz space Lπ and give a generalization of Soria's result [S1]. Let π (t) be a concave function with some nice properties and . If there exists a positive constant a₀ < 1 such that then we have .     

Orlicz空间中三角多项式倒数对周期可微函数的逼近定理

利用Orlicz空间内有关不等式技巧在Orlicz空间内研究了用三角多项式的倒数逼近周期可微函数的问题.得到了一个逼近定理及其推论.     

On a Class of Generalized Lacunary Difference Sequence Spaces Defined by Orlicz Functions

In this article we introduce the generalized lacunary difference sequence spaces [N_θ,M,Δ~m]_0,[N_θ,M,Δ~m]_1 and [N_θ,M,Δ~m]_∞using m~(th)- difference.We study their properties like completeness,solidness,symmetricity.Also we obtain some inclusion relations involving the spaces [N_θ,M,Δ~m]_0,[N_θ,M,Δ~m]_1 and[N_θ,M,Δ~m]_∞and the Cesàro summable and strongly Cesàro summable sequences.     

Orlicz空间的一组不等式

本文将LL~p空间的特征不等式部分地推广到了赋Orlicz范数的Orlicz空间L_M~*中,运用Orlicz空间理论的方法,获得了Orlicz空间L_M~*的一组不等式.     

Remotal Sets in Orlicz Spaces

Let X be a Banach space, (I, μ) be a finite measure space. By L ^Φ(I, X), let us denote the space of all X-valued Bochner Orlicz integrable functions on the unit interval I equipped with the Luxemburg norm. A closed bounded subset G of X is called remotal if for any x ∈ X, there exists g ∈ G such that ‖x ? g‖ = ρ(x, G) = sup {‖x ? y‖: y ∈ G}. In this article, we show that for a separable remotal set G ? X, the set of Bochner integrable functions, L ^Φ(I, G) is remotal in L ^Φ(I, X). Some other results are presented.     

弱紧集与Orlicz空间

本文圆满完善了[1]的结果,并由此给出了Ｏｒｌｉｃｚ空间自反性与Ｓｈｕｒ定理的新证明。     

Parabolic Equations in Orlicz Spaces

An approximation theorem in inhomogeneous Orlicz–Sobolevspaces is proved which allows a second-order parabolic equationin Orlicz spaces to be solved. A trace result is also givenwhich shows that the solutions are continuous with respect totime.