Eugene Ostrovsky
The aim of this paper is investigating of Orlicz spaces with exponential function and correspondence Orlicz norm: we introduce some new equivalent norms, obtain the tail characterization, study the product of functions in Orlicz spaces etc. We consider some applications: estimation of operators in Orlicz spaces and problem of martingales convergence and divergence
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We prove in this article the generalizations on the exponential Orlicz spaces Markov's - Bernstein's inequalities for algebraic polynomials and rational functions.
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.
We derive in this preprint the exact up to multiplicative constant non-asymptotical estimates for the norms of some non-linear in general case operators, for example, the so-called maximal functional operators, in two probabilistic rearrangement invariant norm: exponential Orlicz and Grand Lebesgue Spaces. We will use also the theory of the so-called Grand Lebesgue Spaces (GLS) of measurable functions.
In this article we investigate the Fourier series and transforms for the functions defined on the [-pi, pi]^ d or on the R^d and belonging to the (Bilateral) Grand Lebesgue Spaces. As a particular case we obtain some results about Fourier's transform in the so-called exponential Orlicz spaces. We construct also several examples to show the exactness of offered estimations.
In this paper the necessary and sufficient conditions were given for Orlicz-Lorentz function space endowed with the Orlicz norm having non-squareness and local uniform non-squareness.
Given a normalized Orlicz function $M$ we provide an easy formula for a distribution such that, if $X$ is a random variable distributed accordingly and $X_1,...,X_n$ are independent copies of $X$, then the expected value of the p-norm of the vector $(x_iX_i)_{i=1}^n$ is of the order $\| x \|_M$ (up to constants dependent on p only). In case $p=2$ we need the function $t\mapsto tM'(t) - M(t)$ to be 2-concave and as an application immediately obtain an embedding of the corresponding Orlicz spaces into $L_1[0,1]$. We also provide a general result replacing the $\ell_p$-norm by an arbitrary $N$-no…
In this paper, we present a characterization of support functionals and smooth points in $L_{0}^Φ$, the Musielak-Orlicz space equipped with the Orlicz norm. As a result, criterion for the smoothness of $L_{0}^Φ$ is also obtained. Some expressions involving the norms of functionals in $(L_{0}^Φ)^{*}$, the topological dual of $L_{0}^Φ$, are proved for arbitrary Musielak-Orlicz functions.
Strongly exposed points in Orlicz-Lorentz spaces equipped with the Orlicz norm
本文给出了赋Orlicz范数的Orlicz-Lorentz空间中单位球点为强暴露点的判别准则、空间具有强暴露性质的充要条件，以及处理递减重排相关问题的方法。
Convergence analysis of max-product and max-min Durrmeyer-type exponential sampling operators in Mellin Orlicz space
本文研究了max-product和max-min Durrmeyer型指数采样算子在Mellin Orlicz空间中的收敛性，并通过数值和图形分析验证了其收敛速率和逼近性能。
We provide explicit formulas for the norm of bounded linear functionals on Orlicz-Lorentz function spaces $Λ_{φ,w}$ equipped with two standard Luxemburg and Orlicz norms. Any bounded linear functional is a sum of regular and singular functionals, and we show that the norm of a singular functional is the same regardless of the norm in the space, while the formulas of the norm of general functionals are different for the Luxemburg and Orlicz norm. The relationship between equivalent definitions of the modular $P_{φ,w}$ generating the dual space to Orlicz-Lorentz space is discussed in order to co…
In Orlicz spaces generated by convex Orlicz functions a family of norms generated by some lattice norms in $\mathbb{R}^{2}$ are defined and studied. This family of norms includes the family of the p-Amemiya norms ($1\leq p\leq\infty$) studied in [10-11], [14-15] and [20]. Criteria for strict monotonicity, lower and upper local uniform monotonicities and uniform monotonicities of Orlicz spaces and their subspaces of order continuous elements, equipped with these norms, are given in terms of the generating Orlicz functions, and the lattice norm in $\mathbb{R}^{2}$. The problems of strict convexi…
Noting the important role the abstract $L^p$ space has played in the development of random normed modules, in this paper we introduce and study the Orlicz space generated from a random normed module. First, we give a basic dual space representation theorem which identify the dual of the Orlicz heart of a random normed module with the Orlicz space generated from the random conjugate space. Then, we establish the respective equivalence relations of the strict convexity and uniform convexity of this abstract Orlicz space to the random strict convexity and random uniform convexity of the underlyin…
A dual representation theorem on the conditional Orlicz space generated from a random normed module
本文引入随机Orlicz函数并构造由随机赋范模生成的条件Orlicz空间，证明Orlicz心的稠密性，进而建立对偶表示定理，推广并改进已有结果。
We prove some results about the super Poincaré inequality (SPI) and its relation to the spectrum of an operator: we show that it can be alternatively written with Orlicz norms instead of L1 norms, and we use this to give an alternative proof that a bound on the bottom of the essential spectrum implies a SPI. Finally, we apply these ideas to give a spectral proof of the log Sobolev inequality for the Gaussian measure.
On Orlicz spaces satisfying the Hoffmann-Jørgensen inequality
本研究基于Talagrand的相关证明，完整刻画了满足Hoffmann-Jørgensen型不等式的Orlicz函数，推广了多类集中不等式与有界性结论。
We study the expected value of support functions of random polytopes in a certain direction, where the random polytope is given by independent random vectors uniformly distributed in an isotropic convex body. All results are obtained by an utterly novel approach, using probabilistic estimates in connection with Orlicz norms that were not used in this connection before.
Antilla, Ball and Perissinaki proved that the squares of coordinate functions in $\ell_p^n$ are negatively correlated. This paper extends their results to balls in generalized Orlicz norms on R^n. From this, the concentration of the Euclidean norm and a form of the Central Limit Theorem for the generalized Orlicz balls is deduced. Also, a counterexample for the square negative correlation hypothesis for 1-symmetric bodies is given. Currently the CLT is known in full generality for convex bodies (see the paper "Power-law estimates for the central limit theorem for convex sets" by B. Klartag), w…
Relevant sampling of non-decaying signals in Orlicz and mixed-norm Orlicz spaces defined on a locally compact group
本文针对局部紧群上非衰减信号，在加权Orlicz及混合范数Orlicz空间中建立确定性、平均及随机采样定理，并应用于平移不变空间与Orlicz调制空间。
Convergence Analysis of Max-Min Exponential Neural Network Operators in Orlicz Space
本文提出极大-极小指数神经网络算子逼近函数，扩展出Kantorovich型算子，利用连续性对数模分析逐点与一致收敛速率，并在Orlicz空间考察其收敛行为，通过图形展示逼近误差。
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.
On Subgradients of Convex Functions and Orlicz Pseudo-Norms for Vector-Valued Functions
研究多变量凸函数的可测子梯度构造变种及Δ₂条件的刻画，探讨向量值函数的Luxemburg和Orlicz伪范数基本性质。
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.
We prove in this article that every Borelian measure, for example, the distribution of a random variable, in separable Banach space has a support which is compact embedded Banach subspace; and prove that if the norm of the random variable belongs to some exponential Orlicz space, then the new subspace can be choose such that the norm of this variable in the new space also belongs to other exponential Orlicz space.
In this article we develop the theory of $H$-Orlicz space generated by generalised Young function. Modular convergence of $H$-Orlicz space for the case of vector-valued functions and norm convergence in $\mcH^θ(X, \barμ)$ where $X$ is any Banach space are discussed. Relationships of modular convergence and norm convergence of $H$-Orlicz spaces are discussed.
We give sufficient conditions for the continuity in norm of the translation operator in the Musielak-Orlicz LM spaces. An application to the convergence in norm of approximate identities is given, whereby we prove density results of the smooth functions in LM, in both modular and norm topologies. These density results are then applied to obtain basic topological properties.
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.
In this paper we show that, using combinatorial inequalities and Matrix-Averages, we can generate Musielak-Orlicz spaces, i.e., we prove that $1/π\sum_π \max\limits_{1 \leq i \leq n} \abs{x_i y_{iπ(i)}} \sim \norm{x}_{ΣM_i}$, where the Orlicz functions $M_1,...,M_n$ depend on the matrix $(y_{ij})_{i,j=1}^n$. We also provide an approximation result for Musielak-Orlicz norms which already in the case of Orlicz spaces turned out to be very useful.
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.
Convergence of generalized Orlicz norms with lower growth rate tending to infinity
本文研究了下增长率趋于无穷大时广义Orlicz能量的收敛性，推广了已有结果并放宽假设，使其能处理具有不规则边界和非加倍能量的无界域。
In this article we investigate the Fourier series and transforms for the functions defined on the $ [0, 2 π]^ d $ or $ R^d $ and belonging to the exponential Orlicz and some other rearrangement invariant (r.i.) spaces.
A subspace $H$ of a rearrangement invariant space $X$ on $[0,1]$ is strongly embedded in $X$ if, in $H$, convergence in $X$-norm is equivalent to convergence in measure. We obtain necessary and sufficient conditions on an Orlicz function $M$, under which the unit ball of an arbitrary strongly embedded subspace in the Orlicz space $L_M$ has equi-absolutely continuous norms in $L_M$.
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.
We show that among all Musielak-Orlicz function spaces on a $σ$-finite non-atomic complete measure space equipped with either the Luxemburg norm or the Orlicz norm the only spaces with the Daugavet property are $L_1$, $L_{\infty}$, $L_1\oplus_1 L_{\infty}$ and $L_1\oplus_{\infty} L_{\infty}$. We obtain in particular complete characterizations of the Daugavet property in the weighted interpolation spaces, the variable exponent Lebesgue spaces (Nakano spaces) and the Orlicz spaces.
In this paper we discuss the structure of Orlicz spaces and weak Orlicz spaces on $\mathbb{R}^n$. We obtain some necessary and sufficient conditions for the inclusion property of these spaces. One of the keys is to compute the norm of the characteristic functions of the balls in $\mathbb{R}^n$.
Several local geometric properties of Orlicz space $L_ϕ$ are presented for an increasing Orlicz function $ϕ$ which is not necessarily convex, and thus $L_ϕ$ does not need to be a Banach space. In addition to monotonicity of $ϕ$ it is supposed that $ϕ(u^{1/p})$ is convex for some $p>0$ which is equivalent to that its lower Matuszewska-Orlicz index $α_ϕ>0$. Such spaces are locally bounded and are equipped with natural quasi-norms. Therefore many local geometric properties typical for Banach spaces can also be studied in those spaces. The techniques however have to be different, since duality the…
Sampling discretization in Orlicz spaces
本文研究了有限维空间上Orlicz范数下的抽样离散化问题，获得了新结果，并将其应用于分析不同Orlicz范数下的抽样恢复问题，特别是接近$L^2$范数的线性方法恢复。
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…
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…
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.
Dunkl-Hausdorff operator on weighted Orlicz spaces
本文在加权Orlicz空间中刻画了作用于非负非增函数的Dunkl-Hausdorff算子的模不等式与范数不等式，揭示了该算子的有界性特征。
Certain inequalities between the values of the modular and the norm in the Orlicz spaces are established. These inequalities are applied then to the theory of solvability of nonlinear integral equations of Hammerstein type.
BMO and exponential Orlicz space estimates of the discrepancy function in arbitrary dimension
本文利用二进制调和分析，证明了阶为2的数字网格的偏差函数在BMO^d和exp(L^{2/(d-1)})空间中的范数上限为(log N)^{(d-1)/2}，为理解高维偏差函数的L_∞渐进行为提供了中间步骤。
We show that the weighted Bergman-Orlicz space $A\_α^ψ$ coincides with some weighted Banach space of holomorphic functions if and only if the Orlicz function $ψ$ satisfies the so-called $Δ^{2}$--condition. In addition we prove that this condition characterizes those $A\_α^ψ$ on which every composition operator is bounded or order bounded into the Orlicz space $L\_α^ψ$. This provides us with estimates of the norm and the essential norm of composition operators on such spaces. We also prove that when $ψ$ satisfies the $Δ^{2}$--condition, a composition operator is compact on $A\_α^ψ$ if and only …
Suppose that a real nonatomic function space on $[0,1]$ is equipped with two re\-arran\-ge\-ment-invariant norms $\|\cdot\|$ and $|||\cdot|||$. We study the question whether or not the fact that $(X,\|\cdot\|)$ is isometric to $(X,|||\cdot|||)$ implies that $\|f\|= |||f|||$ for all $f$ in $X$. We show that in strictly monotone Orlicz and Lorentz spaces this is equivalent to asking whether or not the norms are defined by equal Orlicz functions, resp. Lorentz weights. We show that the above implication holds true in most rearrangement-invariant spaces, but we also identify a class of Orlicz spac…
On Köthe duals of Orlicz-Lorentz spaces
本文研究Orlicz-Lorentz空间的Köthe对偶空间$\u200b\mathcal{M}_{φ,w}$的性质，刻画其序连续子空间、可分性与Radon-Nikodým性质，给出空间比较关系，并应用于M-嵌入子空间与保范延拓问题。
Some necessary and sufficient conditions are found for Banach function lattices to have the Radon-Nikodým property. Consequently it is shown that an Orlicz space $L_φ$ over a non-atomic $σ$-finite measure space $(Ω, Σ,μ)$, not necessarily separable, has the Radon-Nikodým property if and only if $φ$ is an $N$-function at infinity and satisfies the appropriate $Δ_2$ condition. For an Orlicz sequence space $\ell_φ$, it has the Radon-Nikodým property if and only if $φ$ satisfies condition $Δ_2^0$. In the second part the relationships between uniformly $\ell_1^2$ points of the unit sphere of a Bana…
For an Orlicz function $φ$ and a decreasing weight $w$, two intrinsic exact descriptions are presented for the norm in the Köthe dual of an Orlicz-Lorentz function space $Λ_{φ,w}$ or a sequence space $λ_{φ,w}$, equipped with either Luxemburg or Amemiya norms. The first description of the dual norm is given via the modular $\inf\{\intφ_*(f^*/|g|)|g|: g\prec w\}$, where $f^*$ is the decreasing rearrangement of $f$, $g\prec w$ denotes the submajorization of $g$ by $w$ and $φ_*$ is the complementary function to $φ$. The second one is stated in terms of the modular $\int_I φ_*((f^*)^0/w)w$, where $…
In this paper we present sufficient and necessary conditions for inclusion relation between two weighted Orlicz spaces which complete the Osançliol result in 2014. One of the keys to prove our results is to use the norm of the characteristic functions of the balls in $\mathbb{R}^n$.
Orlicz-Lorentz spaces provide a common generalization of Orlicz spaces and Lorentz spaces. They have been studied by many authors, including Mastyło, Maligranda, and Kamińska. In this paper, we consider the problem of comparing the Orlicz-Lorentz norms, and establish necessary and sufficient conditions for them to be equivalent. As a corollary, we give necessary and sufficient conditions for a Lorentz-Sharpley space to be equivalent to an Orlicz space, extending results of Lorentz and Raynaud. We also give an example of a rearrangement invariant space that is not an Orlicz-Lorentz space.
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 define bicomplex Orlicz space with hyperbolic valued Luxemburg norm and discussed some of their properties. We have also partially characterize an integral representation of a $\mathbb{D}$-valued convex function. Further we have shown that the spectrum of the unilateral shift operator on $l^2(\mathbb{BC})$ is the null cone $\mathcal{NC}$.
Heat equations associated to harmonic oscillator with exponential nonlinearity
研究涉及分数阶调和振子和指数非线性的热方程 Cauchy 问题，在 Orlicz 空间中建立局部适定性，证明小初始数据下全局弱-温和解的存在性，给出大时间衰减估计，并指出某些非负初始数据下局部非负经典解可能不存在。
In this paper we consider composition operator $C_φ generated by nonsingular measurable transformation $T$ and multiplication operator $M_u$ generated by measurable function $u$ between two different Orlicz spaces, then we investigate boundedness, compactness and essential norm of multiplication and composition operators in term of properties of the mapping $φ$, the function $u$ and the measure space $(X, Σ, μ)$.
In this paper we discuss the structure of weighted weak Lebesgue spaces and weighted weak Orlicz spaces on $\mathbb{R}^n$. First, we present sufficient and necessary conditions for inclusion relation between weighted weak Lebesgue spaces. Next, we also obtain similar results on weighted weak Orlicz spaces. One of the keys to prove our results is to use the norm of the characteristic functions of the balls in $\mathbb{R}^n$.
The norm in classical Sobolev spaces can be expressed as a difference quotient. This expression can be used to generalize the space to the fractional smoothness case. Because the difference quotient is based on shifting the function, it cannot be used in generalized Orlicz spaces. In its place, we introduce a smoothed difference quotient and show that it can be used to characterize the generalized Orlicz-Sobolev space. Our results are new even in Orlicz spaces and variable exponent spaces.
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.
We discuss the optimality of Cwikel-Solomyak estimates for the uniform operator norm and establish optimality of M.Z Solomyak's results \cite{Solomyak1995} within the class of Orlicz spaces. Our methods are based on finding the optimal version of the Sobolev embedding theorem.
Inclusion relations among fractional Orlicz-Sobolev spaces and a Littlewood-Paley characterization
本文刻画了不同光滑性分数阶 Orlicz-Sobolev 空间的嵌入关系，并利用新的 Orlicz 卷积最优不等式建立了 Gagliardo-Slobodeckij 范数与 Littlewood-Paley 分解等范数的等价性，为嵌入证明提供关键工具。
There are two versions of Orlicz-Morrey spaces (on $\mathbb{R}^n$), defined by Nakai in 2004 and by Sawano, Sugano, and Tanaka in 2012. In this paper we discuss the inclusion properties of these two spaces and compare the results. Computing the norms of the characteristic functions of balls in $\mathbb{R}^n$ is one of the keys to our results. Similar results for weak Orlicz-Morrey spaces of both versions are also obtained.
The Hardy averaging operator between Orlicz spaces and a new gauge functional characterizing a given Orlicz space
该研究证明Hardy平均算子的最优Orlicz定义域与最优重排不变定义域重合，提出新泛函提升对偶范数计算性，还得到Hardy-Littlewood极大函数等算子的最优Orlicz空间映射性质。
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