Denny H. Leung
Let $M$ be a non-degenerate Orlicz function such that there exist $\ep > 0$ and $0 < s < 1$ with $\su M(\ep s^i)/M(s^i) < \infty$. It is shown that the Orlicz sequence space $h_M$ is isomorphic to a subspace of $C(\om^\om)$. It is also shown that for any non-degenerate Orlicz function $M$, $h_M$ does not embed into $C(\al)$ for any $\al < \om^\om$.
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We deal with Orlicz-Sobolev embeddings in open subsets of $\mathbb{R}^n$. A necessary and sufficient condition is established for the existence of an optimal, i.e. largest possible, Orlicz-Sobolev space continuously embedded into a given Orlicz space. Moreover, the optimal Orlicz-Sobolev space is exhibited whenever it exists. Parallel questions are addressed for Orlicz-Sobolev embeddings into Orlicz spaces with respect to a Frostman measure, and, in particular, for trace embeddings on the boundary.
The optimal Orlicz target space is exhibited for embeddings of fractional-order Orlicz-Sobolev spaces in $\mathbb R^n$. An improved embedding with an Orlicz-Lorentz target space, which is optimal in the broader class of all rearrangement-invariant spaces, is also established. Both spaces of order $s\in (0,1)$, and higher-order spaces are considered. Related Hardy type inequalities are proposed as well. An extension theorem is proved, that enables us to derive embeddings for spaces defined in Lipschitz domains. Necessary and sufficient conditions for the compactness of fractional Orlicz-Sobolev…
We provide a necessary condition on the regularity of domains for the optimal embeddings of first order (and higher order) Orlicz-Sobolev spaces into Orlicz spaces in the sense of \cite{Cia96} (and \cite{Cia06}).
In [K--S 1] it was shown that $$ \underset π \to {\text{Ave}} (\sum_{i=1}^{n}|x_i a_{π(i)}|^2)^{\frac {1}{2}} $$ is equivalent to an Orlicz norm whose Orlicz function is 2-concave. Here we give a formula for the sequence $a_1, a_2,....,a_n$ so that the above expression is equivalent to a given Orlicz norm.
We give a sufficient (and, in the case of a compact domain, a necessary) condition for the embedding of Sobolev space of functions with integrable gradient into Besov-Orlicz spaces to be bounded. The condition has a form of a simple integral inequality involving Young and weight functions. We provide an example with Matuszewska-Orlicz indices of involved Orlicz norm equal to one. The main tool is the molecular decomposition of functions from a BV space.
In this paper we present a method for determining whether there exists a largest Orlicz space $L^A(Ω)$ satisfying the Sobolev embedding $W^mL^A(Ω) \to Y(Ω)$ where $Y(Ω)$ stands for an arbitrary so-called Marcinkiewicz endpoint space. The tool developed in this work enables us to investigate the optimality of Orlicz domain spaces in Sobolev embeddings and also in Sobolev trace embeddings on domains $Ω$ in $\mathbb{R}^n$ with various regularity.
On the Continuous embeddings between the fractional Hajłasz-Orlicz-Sobolev spaces
本文在Orlicz函数满足特定条件时，证明了度量测度空间上分数阶Orlicz-Sobolev空间与分数阶Hajłasz-Orlicz-Sobolev空间之间的连续嵌入关系。
We give an almost complete description of the coarse and uniform embeddability between Orlicz sequence spaces. We show that the embeddability between two Orlicz sequence spaces is in most cases determined only by the values of their upper Matuszewska-Orlicz indices. On the other hand, we present examples which show that sometimes the embeddability is not determined by the values of these indices.
Fractional Orlicz-Sobolev embeddings into Campanato type spaces
该研究刻画了欧氏空间上分数阶Orlicz-Sobolev空间到各类Campanato型空间的最优嵌入，推导了BMO、VMO相关特例并指出其与经典分数阶Sobolev空间嵌入的差异。
Musielak-Orlicz-Sobolev embeddings: Necessary and Sufficient Conditions
该研究针对形如$Φ(x,t)=t^{p(x)}(\log(e+t))^{q(x)}$的Musielak-Orlicz-Sobolev空间，探究其嵌入性质关于区域的充要条件。
On the compactness of embeddings for a class of weighted Orlicz-Sobolev sequence spaces
本文引入一类加权Orlicz-Sobolev序列空间，证明其连续性与紧性准则，连续性源于Orlicz函数逐点比较，紧性结合Schauder基与局部条件，并给出实例与开放问题。
We characterize norm one complemented subspaces of Orlicz sequence spaces $\ell_M$ equipped with either Luxemburg or Orlicz norm, provided that the Orlicz function $M$ is sufficiently smooth and sufficiently different from the square function. This paper concentrates on the more difficult real case, the complex case follows from previously known results.
The purpose of this paper is to introduce the space of geometric sequences that are strongly summable with respect to an Orlicz function and the Fibonacci difference sequences.Also some topological properties and inclusion relations between the resulting geometric sequence spaces are discussed here.
In this paper we introduced the generalized difference Cesaro sequence spaces of fuzzy real numbers defined by Orlicz function. We study some topological properties of these spaces. We obtain some inclusion relations involving these sequence spaces. These notions generalize many notions on difference Cesaro sequence spaces.
A necessary and sufficient condition for fractional Orlicz-Sobolev spaces to be continuously embedded into $L^\infty(\mathbb R^n)$ is exhibited. Under the same assumption, any function from the relevant fractional-order spaces is shown to be continuous. Improvements of this result are also offered. They provide the optimal Orlicz target space, and the optimal rearrangement-invariant target space in the embedding in question. These results complement those already available in the subcritical case, where the embedding into $L^\infty(\mathbb R^n)$ fails. They also augment a classical embedding t…
We study the complex interpolation and derivation process induced by a family of Orlicz sequence spaces. We present a concrete example of interpolation family of three spaces inducing a centralizer that cannot be obtained from complex interpolation of two spaces.
We show that if $(\ell_{ϕ_0}, \ell_{ϕ_1})$ is a couple of suitable Orlicz sequence spaces then the corresponding Rochberg derived spaces of all orders associated to the complex interpolation method are Fenchel-Orlicz spaces. In particular, the induced twisted sums have the $(C[0, 1], \mathbb{C})$-extension property.
We study point-wise estimates for the modified Riesz potential. We show that the point-wise estimates imply embeddings into Orlicz spaces from the L^1_p-space where the functions are defined in non-smooth domains. The Orlicz functions depend on the geometry of the domain. We show that the Orlicz function is optimal when p=1.
Perturbation Method in Orlicz Sequence Spaces
本文在满足零点$Δ_2$条件的Orlicz序列空间中提出新的扰动方法，通过扰动定义函数$σ_M$支撑有界下下半连续函数，并举例说明其可用于判断Orlicz空间的光滑性类型。
Almost compact embeddings between Orlicz and Lorentz spaces
本文利用涉及 p,q 和 Young 函数 A 的平衡条件，刻画了 Orlicz 空间 L^A 到 Lorentz 空间 L^{p,q} 的几乎紧嵌入，并提出一种基于水平集测度的 Young 型不等式新方法。
We prove that the set of all integrable functions whose sequences of negative (resp. nonnegative) Fourier coefficients belong to $\ell^1\cap\ell^Φ_{ϕ,w}$ (resp. to $\ell^1\cap\ell^Ψ_{ψ,\varrho}$), where $\ell^Φ_{ϕ,w}$ and $\ell^Ψ_{ψ,\varrho}$ are two-weighted Orlicz sequence spaces, forms an algebra under pointwise multiplication whenever the weight sequences \[ ϕ=\{ϕ_n\},\quad ψ=\{ψ_n\},\quad w=\{w_n\},\quad \varrho=\{\varrho_n\} \] increase and satisfy the $Δ_2$-condition.
In this article, we show that Orlicz-Lorentz spaces $\ell^n_{M,a}$, $n\in\mathbb N$ with Orlicz function $M$ and weight sequence $a$ are uniformly isomorphic to subspaces of $L_1$ if the norm $\|\cdot\|_{M,a}$ satisfies certain Hardy-type inequalities. This includes the embedding of some Lorentz spaces $d^n(a,p)$. Our approach is based on combinatorial averaging techniques and we prove a new result of independent interest that relates suitable averages with Orlicz-Lorentz norms.
We study some new strongly almost lacunary statistical $A$-convergent sequence space of order $α$ defined by a Musielak-Orlicz function. We also give some inclusion relations between the newly introduced class of sequences with the spaces of strongly almost lacunary $A$-convergent sequence of order $α$. Moreover we have examined some results on Musielak-Orlicz function with respect to these spaces.
Isometric rigidity and Fraïssé properties of Orlicz sequence spaces
论文研究了Orlicz序列空间的近等距刚性性质，证明了某些空间guarded Fraïssé但非ω-范畴，且不含ℓ₂的拷贝，回答了Cúth-de Rancourt-Doucha的问题。
The ergodicity of Orlicz sequence spaces
本文证明非希尔伯特可分Orlicz序列空间具有遍历性，含连续统多个两两不同构子空间，同时推导Kalton-Peck构造的相关扭曲希尔伯特空间要么为希尔伯特空间要么具有遍历性。
Let (f_n) and (g_n) be two sequences of random variables adapted to an increasing sequence of $σ$-algebras $({\cal F}_n)$ such that the conditional distributions of f_n and g_n given ${\cal F}_{n-1}$ coincide, and such that the sequence (g_n) is conditionally independent. Then it is known that $\normo{\sum f_k}_p \le C \normo{\sum g_k}_p$, $1 \le p \le \infty$ where the constant C is independent of p. The aim of this paper is to extend this result to certain classes of Orlicz and rearrangement invariant spaces. This paper includes fairly general techniques for obtaining rearrangement invariant…
By a stronger compact boundary embedding theorem in Musielak-Orlicz-Sobolev space developed in the paper, variational method is employed to deal with the nonlinear elliptic equation with the nonlinear Neumann boundary condition in the framework of Musielak-Orlicz-Sobolev space.
Optimality of embeddings in Orlicz spaces
本文研究欧氏域Mazya类中Orlicz-Sobolev嵌入的最优性问题，证明了临界状态下部分此类嵌入不存在最优Orlicz空间，其关键特例为John域上的Brezis-Wainger嵌入。
Let $\boldΦ=(ϕ_n)$ be a Musielak-Orlicz function, $X$ be a real Banach space and $A$ be any infinite matrix. In this paper, a generalized vector-valued Musielak-Orlicz sequence space $l_{\bold Φ}^{A}(X)$ is introduced. It is shown that the space is complete normed linear space under certain conditions on the matrix $A$. It is also shown that $l_{\boldΦ}^{A}(X)$ is a $σ$- Dedikind complete whenever $X$ is so. We have discussed some geometric properties, namely, uniformly monotone, uniform Opial property for this space. Using the sequence of $s$-number (in the sense of Pietsch), the operators of…
A Banach space is polyhedral if the unit ball of each of its finite dimensional subspaces is a polyhedron. It is known that a polyhedral Banach space has a separable dual and is $c_0$-saturated, i.e., each closed infinite dimensional subspace contains an isomorph of $c_0$. In this paper, we show that the Orlicz sequence space $h_M$ is isomorphic to a polyhedral Banach space if $\lim_{t\to 0}M(Kt)/M(t) = \infty$ for some $K < \infty$. We also construct an Orlicz sequence space $h_M$ which is $c_0$-saturated, but which is not isomorphic to any polyhedral Banach space. This shows that being $c_0$…
We consider a sequence of positive smooth critical points of the Adams-Moser-Trudinger embedding of $H^m_0$ into Orlicz spaces. We study its concentration-compactness behavior and show that if the sequence is not precompact, then the liminf of the $H^m_0$-norms of the functions is greater than or equal to a positive geometric constant.
Carleson embeddings and pointwise multipliers between Hardy-Orlicz spaces and Bergman-Orlicz spaces of the upper half-plane
本文给出涉及凹或凸增长函数的 Carleson 测度的一般刻画，并据此建立上半平面 Hardy-Orlicz 空间与 Bergman-Orlicz 空间之间的连续嵌入与逐点乘子集合的刻画。
Let $M_1$ and $M_2$ be functions on $[0,1]$ such that $M_1(t^{1/p})$ and $M_2(t^{1/p})$ are Orlicz functions for some $p \in (0,1].$ Assume that $M_2^{-1} (1/t)/M_1^{-1} (1/t)$ is non-decreasing for $t \geq 1.$ Let $(α_i)_{i=1}^\infty$ be a non-increasing sequence of non-negative real numbers. Under some conditions on $(α_i)_{i=1}^\infty,$ sharp two-sided estimates for entropy numbers of diagonal operators $T_α:\ell_{M_1} \rightarrow \ell_{M_2}$ generated by $(α_i)_{i=1}^\infty,$ where $\ell_{M_1}$ and $\ell_{M_2}$ are Orlicz sequence spaces, are proved. The results generalise some works of Ed…
Weak compactness in nice Musielak-Orlicz spaces
本文在满足Δ₂条件的N-函数下，证明了Musielak-Orlicz空间中的弱紧性准则，并展示了该空间具有子序列分裂性质时的弱Banach-Saks性质。
For a Young function $ϕ$ and a locally compact second countable group $G,$ let $L^ϕ(G)$ denote the Orlicz space on $G.$ In this article, we present a necessary and sufficient condition for the topological transitivity of a sequence of cosine operators $\{C_n\}_{n=1}^{\infty}:=\{\frac{1}{2}(T^n_{g,w}+S^n_{g,w})\}_{n=1}^{\infty}$, defined on $L^ϕ(G)$. We investigate the conditions for a sequence of cosine operators to be topological mixing. Moreover, we go on to prove the similar results for the direct sum of a sequence of the cosine operators. At the last, an example of a topological transitive…
Perturbation Method in Musielak-Orlicz Sequence Spaces
本文推广了Banach空间中的抽象变分原理，证明了扰动集不仅包含稠密Gδ子集，还补全了特定正锥的σ-孔集。同时，若空间为满足ℓ_Φ≈h_Φ的Musielak-Orlicz序列空间，则WPMC概念被更强的Tikhonov良好性替代。
On the modulus of continuity of fractional Orlicz-Sobolev functions
该研究给出分数阶Orlicz-Sobolev空间连续嵌入一致连续函数空间的充要条件与最优连续模，补充超临界Sobolev情形的嵌入结果，证明需采用新策略。
In this note we prove that $\frac{1}{n!} \sum_π (\sum_{i=1}^n |x_i a_{i,π(i)} |^2)^{1/2}$ is equivalent to a Musielak-Orlicz norm $\norm{x}_{\sum M_i}$. We also obtain the inverse result, i.e., given the Orlicz functions, we provide a formula for the choice of the matrix that generates the corresponding Musielak-Orlicz norm. As a consequence, we obtain the embedding of 2-concave Musielak-Orlicz spaces into L_1.
We provide necessary and sufficient conditions for the coincidence, up to equivalence of the norms, between strong and weak Orlicz spaces. Roughly speaking, this coincidence holds true only for the so-called exponential spaces. We find also the exact value of the embedding constant which appears in the corresponding norm inequality.
Unconditional basic sequences in function spaces with applications to Orlicz spaces
本文研究了函数空间 $\bf{L}$ 的无条件基序列，提出了一个二分法条件，使得其互补子空间的无条件基要么等价于 $\ell_2$ 的单位向量系，要么具有不相交支撑的子基，并据此分类了对称基序列，应用于 Orlicz 空间。
In this paper, the definition of noncommutative Orlicz sequence spaces is given, these spaces generalize the Schatten classes Sp(H). After some relations of trace and norm on this spaces have been researched, one give the criterion of reflexivity of these spaces. At last, as an application, we find the Toeplitz operator on the Bergman space belongs to some noncommutative Orlicz sequence spaces, hence the trace and the norm of operator could be computed.
Sobolev embeddings in Musielak-Orlicz space
本文提出了基于一般Musielak-Orlicz范数的Sobolev空间嵌入定理，通过引入最佳Sobolev共轭建立了Sobolev及Sobolev-Poincaré不等式，并利用Riesz位势的新型弱型不等式恢复并补充了多种标准空间在临界情况下的最佳嵌入。
We construct an analytic self-map $ϕ$ of the unit disk and an Orlicz function $Ψ$ for which the composition operator of symbol $ϕ$ is compact on the Hardy-Orlicz space $H^Ψ$, but not compact on the Bergman-Orlicz space ${\mathfrak B}^Ψ$. For that, we first prove a Carleson embedding theorem, and then characterize the compactness of composition operators on Bergman-Orlicz spaces, in terms of Carleson function (of order 2). We show that this Carleson function is equivalent to the Nevanlinna counting function of order 2.
Symmetric Bases and $\ell^p$ Embeddings in Musielak--Orlicz Sequence Spaces under $δ_2$
该研究探讨满足δ₂条件的Musielak–Orlicz序列空间ℓ_Φ中的同构嵌入问题，构造Orlicz函数极限族，推导ℓ^p嵌入的充要条件及余型判据，通过实例验证相关界的有效性。
In this paper we study the theory of the so-called Kantorovich max-product neural network operators in the setting of Orlicz spaces $L^φ$. The results here proved, extend those given by Costarelli and Vinti in Result Math., 2016, to a more general context. The main advantage in studying neural network type operators in Orlicz spaces relies in the possibility to approximate not necessarily continuous functions (data) belonging to different function spaces by a unique general approach. Further, in order to derive quantitative estimates in this context, we introduce a suitable K-functional in $L^…
Let $1\le p\le\infty$. A Banach lattice $X$ is said to be $p$-disjointly homogeneous or $(p-DH)$ (resp. restricted $(p-DH)$) if every normalized disjoint sequence in $X$ (resp. every normalized sequence of characteristic functions of disjoint subsets) contains a subsequence equivalent in $X$ to the unit vector basis of $\ell_p$. We revisit $DH$-properties of Orlicz spaces and refine some previous results of this topic, showing that $(p-DH)$-property is not stable in the class of Orlicz spaces and the classes of restricted $(p-DH)$ and $(p-DH)$ Orlicz spaces are different. Moreover, we give a c…
Orlicz-Sobolev embeddings and heat kernel based Besov classes
针对具有不同局部和全局结构的度量测度空间，本文研究Besov空间与有界变差函数相关泛函不等式，并发展$L^1$理论以刻画有限周界集和等周不等式中的结构差异。
Orlicz-Schatten Factorizations for Non-Commutative Sobolev Embeddings
本研究提出通过新定义的Orlicz-Schatten序列理想分解量子环面上非交换Sobolev空间嵌入的框架，建立量子拉普拉斯嵌入的求和算子刻画，证明分解定理的存在性与最优性，并给出其在非交换PDE正则性、量子信息度量中的应用，拓展了相关结构在泛函分析与数学物理中的应用范围。
We prove asymptotic formulas for Toeplitz determinants generated by functions with sequences of Fourier coefficients belonging to weighted Orlicz sequence classes. We concentrate our attention on the case of nonvanishing generating functions with nonzero Cauchy index.
We prove a bi-sublinear embedding for semigroups generated by non-smooth complex-coefficient elliptic operators in divergence form and for certain mutually dual pairs of Orlicz-space norms. This generalizes a result by Carbonaro and Dragičević from power functions to more general Young functions that still behave like powers. To achieve this, we generalize a Bellman function constructed by Nazarov and Treil.
Characterizations are given for 1-complemented hyperplanes of strictly monotone real Lorentz spaces and 1-complemented finite codimensional subspaces (which contain at least one basis element) of real Orlicz spaces equipped with either Luxemburg or Orlicz norm.
Let $\mathbb{Y}$ be either an Orlicz sequence space or a Marcinkiewicz sequence space. We take advantage of the recent advances in the theory of factorization of the identity carried on in [R. Lechner, Subsymmetric weak* Schauder bases and factorization of the identity, arXiv:1804.01372 [math.FA]] to provide conditions on $\mathbb{Y}$ that ensure that, for any $1\le p\le\infty$, the infinite direct sum of $\mathbb{Y}$ in the sense of $\ell_p$ is a primary Banach space, enlarging this way the list of Banach spaces that are known to be primary.
Convergence results in Orlicz spaces for sequences of max-product Kantorovich sampling operators
该研究建立了Orlicz空间中基于广义核的max-product型Kantorovich采样算子的模收敛统一理论，覆盖有界区间与实轴函数，适用于多种函数空间的非连续信号重构并给出应用示例。
Operators on Orlicz sequence spaces and $Δ_2$-fundamentality
研究离散极大算子和平均算子在Orlicz空间上有界的刻画条件，引入$\Delta_2$-基础序列概念，给出仅涉及相邻项比值上极限的简单刻画。
Let G be a discrete group. Suppose that the reduced group C*-algebra of G is simple. We use results of Kalantar-Kennedy and Haagerup, and Banach space interpolation, to prove that, for p in (1,infinity), the reduced group L^p operator algebra F^p_r(G) and its *-analog B^{p,*}_r(G) are simple. If G is countable, we prove that the Banach algebras generated by the left regular representations on reflexive Orlicz sequence spaces and certain Lorentz sequence spaces are also simple. We prove analogous results with simplicity replaced by the unique trace property. For use in the Orlicz sequence space…
Given an Orlicz function $M$, we show which random variables $ξ_i$, $i=1,...,n$ generate the associated Orlicz norm, i.e., which random variables yield $\mathbb{E} \max\limits_{1\leq i \leq n}|x_iξ_i| \sim \norm{(x_i)_{i=1}^n}_M$. As a corollary we obtain a representation for the distribution function in terms of $M$ and $M'$ which can be easily applied to many examples of interest.
Let $\Ps(\N)$ be the set of all finite subsets of $\N$, endowed with the product topology. A description of the compact subsets of $\Ps(\N)$ is given. Two applications of this result to Banach space theory are shown : (1) a characterization of the symmetric sequence spaces which embed into $C(\om^\om)$, and (2) a characterization, in terms of the Orlicz function $M$, of the Orlicz sequence spaces $h_M$ which embed into $C(K)$ for some countable compact Hausdorff space $K$.
An old problem of P. Levy is to characterize those Banach spaces which embed isometrically in $L_p.$ We show a new criterion in terms of the second derivative of the norm. As an application, we show that if $M$ is a twice differentiable Orlicz function with $M'(0)=M''(0)=0$ then the $n$-dimensional Orlicz space $\ell_M^n,\ n\ge 3,$ does not embed isometrically in $L_p$ with $0<p\le 2.$ These results generalize and clear up the recent solution to the 1938 Schoenberg's problem on positive definite functions.
Factoring the Sobolev embedding operator
本文通过两种方法研究Sobolev嵌入算子的分解与求和性质，包括基于Young函数和Marcinkiewicz采样定理的分解，以及基于$(p,1)$-求和范数的外推方法。
Necessary and sufficient conditions under which the Cesàro--Orlicz sequence space $\cfi$ is nontrivial are presented. It is proved that for the Luxemburg norm, Cesàro--Orlicz spaces $\cfi$ have the Fatou property. Consequently, the spaces are complete. It is also proved that the subspace of order continuous elements in $\cfi$ can be defined in two ways. Finally, criteria for strict monotonicity, uniform monotonicity and rotundity (= strict convexity) of the spaces $\cfi$ are given.
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