Orlicz Classes Research Articles

Norms of sub-exponential random vectors

We discuss various forms of the Luxemburg norm in spaces of random vectors with coordinates belonging to the classical Orlicz spaces of exponential type. We prove equivalent relations between some kinds of these forms. We also show when the so-called uniform norm is majorized by norms of coordinates up to some constants. We give an application of other norm to study of chaos in random vectors with sub-exponential coordinates.

Orlicz–Sobolev nematic elastomers

We extend the existence theorems in Barchiesi et al. (2017), for models of nematic elastomers and magnetoelasticity, to a larger class in the scale of Orlicz spaces. These models consider both an elastic term where a polyconvex energy density is composed with an unknown state variable defined in the deformed configuration, and a functional corresponding to the nematic energy (or the exchange and magnetostatic energies in magnetoelasticity) where the energy density is integrated over the deformed configuration. In order to obtain the desired compactness and lower semicontinuity, we show that the regularity requirement that maps create no new surface can still be imposed when the gradients are in an Orlicz class with an integrability just above the space dimension minus one. We prove that the fine properties of orientation-preserving maps satisfying that regularity requirement (namely, being weakly 1-pseudomonotone, H1-continuous, a.e. differentiable, and a.e. locally invertible) are still valid in the Orlicz–Sobolev setting.

On Almost Everywhere Convergence of Tensor Product Spline Projections

Let d∈N, and let f be a function in the Orlicz class L(log+L)d−1 defined on the unit cube [0,1]d in Rd. Given knot sequences Δ1,…,Δd on [0,1], we first prove that the orthogonal projection P(Δ1,…,Δd)(f) onto the space of tensor product splines with arbitrary orders (k1,…,kd) and knots Δ1,…,Δd converges to f almost everywhere as the mesh diameters |Δ1|,…,|Δd| tend to zero. This extends the one-dimensional result in [9] to arbitrary dimensions. In the second step, we show that this result is optimal, that is, given any “bigger” Orlicz class X=σ(L)L(log+L)d−1 with an arbitrary function σ tending to zero at infinity, there exist a function φ∈X and partitions of the unit cube such that the orthogonal projections of φ do not converge almost everywhere.

Entire functions of least deviation from zero in generalized Orlicz classes

Entire functions of least deviation from zero in generalized Orlicz classes

Dilation-commuting operators on power-weighted Orlicz classes

Let $\Phi_1$ and $\Phi_2$ be nondecreasing functions from $\mathbb{R_+}=(0,\infty)$ onto itself. For $i=1,2$ and $\gamma \in \mathbb{R}$, define the Orlicz class $L_{\Phi_{i}}(\mathbb{R_+})$ to be the set of Lebesgue-measurable functions $f$ on $\mathbb{R_+}$ such that \begin{equation*} \int_{\mathbb{R_+}} \Phi_{i} \left( k|(Tf)(t)| \right) t^{\gamma}dt 0$. Our goal in this paper is to find conditions on $\Phi_1$, $\Phi_2$, $\gamma$ and an operator $T$ so that the assertions \begin{equation} T : L_{\Phi_2,t^{\gamma}}(\mathbb{R_+}) \rightarrow L_{\Phi_1,t^{\gamma}}(\mathbb{R_+}), \tag{I} \end{equation} and \begin{equation}\label{modularA} \int_{\mathbb{R_+}} \Phi_1 \left( |(Tf)(t)| \right)t^{\gamma}dt \leq K \int_{\mathbb{R_+}} \Phi_2 \left( K|f(s)| \right)s^{\gamma}ds, \tag{M} \end{equation} in which $K>0$ is independent of $f$, say, simple on $\mathbb{R_+}$, are equivalent and to then find necessary and sufficient conditions in order that (\ref{modularA}) holds.

Normed Orlicz function spaces which can be quasi-renormed with easily calculable quasinorms

We are interested in the widest possible class of Orlicz functions Φ such that the easily calculable quasinorm [f]Φ,p:=‖f‖E{IΦ(f‖f‖E)}1p if f≠0 and [f]Φ,p=0 if f=0, on the Orlicz space LΦ(Ω,Σ,μ) generated by Φ, is equivalent to the Luxemburg norm ‖⋅‖Φ. To do this, we use a suitable Δ2-condition, lower and upper Simonenko indices pSa(Φ) and qSa(Φ) for the generating function Φ, numbers p∈[1,pSa(Φ)] satisfying qSa(Φ)−p≤1, and an embedding of LΦ(Ω,Σ,μ) into a suitable Köthe function space E=E(Ω,Σ,μ). We take as E the Lebesgue spaces Lr(Ω,Σ,μ) with r∈[1,pSl(Φ)], when the measure μ is nonatomic and finite, and the weighted Lebesgue spaces Lωr(Ω,Σ,μ), with r∈[1,pSa(Φ)] and a suitable weight function ω, when the measure μ is nonatomic infinite but σ-finite. We also use condition ∇3 if pSa(Φ)=1 and condition ∇2 if pSa(Φ)>1, proving their necessity in most of the considered cases. Our results seem important for applications of Orlicz function spaces.

The locally uniformly non-square points of Orlicz-Bochner sequence spaces

In this paper, we investigate the locally uniformly non-square point of Orlicz–Bochner sequence spaces endowed with Luxemburg norm. Analysing and combining the generating function M and properties of the real Banach space X, we get sufficient and necessary conditions of locally uniformly non-square point, which contributes to criteria for locally uniform non-squareness in Orlicz–Bochner sequence spaces. The results generalize the corresponding results in the classical Orlicz sequence spaces.

Geometric properties of Orlicz spaces equipped with \(p\)-Amemiya norms − results and open questions

The classical Orlicz and Luxemburg norms generated by an Orlicz function \(\Phi\) can be defined with the use of the Amemiya formula [H. Hudzik and L. Maligranda, Amemiya norm equals Orlicz norm in general, Indag. Math. 11 (2000), no. 4, 573-585]. Moreover, in this article Hudzik and Maligranda suggested investigating a family of p-Amemiya norms defined by the formula \(\|x\|_{\Phi,p}=\inf_{k>0} \frac{1}{k} (1+I_\Phi^p(kx))^{1/p}\), where \(1\le p\le\infty\) (under the convention: \((1+u^\infty)^{1/\infty}=\lim_{p\to\infty}(1+u^p)^{1/p}=\max{1,u}\) for all \(u\ge 0\)). Based on this idea, a number of papers have been published in the past few years. In this paper, we present some major results concerning the geometric properties of Orlicz spaces equipped with p-Amemiya norms. In the last section, a more general case of Amemiya type norms is investigated. A few open questions concerning this theory will be stated as well.

Convergence and localization in Orlicz classes for multiple Walsh–Fourier series with a lacunary sequence of rectangular partial sums

For functions f in Orlicz classes, we consider multiple Walsh–Fourier series for which the rectangular partial sums Sn(x;f) have indices n=(n1,…,nN)∈ZN (N≥3), where either N or N−1 components are elements of (single) lacunary sequences. For this series, we prove the validity of weak generalized localization almost everywhere on an arbitrary measurable set A⊂IN={x∈RN:0≤xj<1,j=1,2,…,N}, in the case when the structure and geometry of A are defined by the properties Bk, 2≤k≤N. We define the relation between the parameter k and the “smoothness” of functions in terms of the Orlicz classes. As a consequence, we obtain some results on the “local smoothness conditions.” In particular, the theorem is proved for the convergence of Walsh–Fourier series on an arbitrary open set Ω⊂IN under the minimal conditions imposed on the smoothness of the function on this set.

Localization for multiple Fourier series with “J k -lacunary sequence of partial sums” in Orlicz classes

We obtain structural and geometric characteristics of sets on which weak generalized localization almost everywhere is valid for multiple trigonometric Fourier series of functions in the classes \(L(\log ^ + L)^{3k + 2} (\mathbb{T}^N ),1 \leqslant k \leqslant N - 2,N \geqslant 3\), in the case where the rectangular partial sums of these series have an “index” in which exactly k components are elements of lacunary sequences.

Packing Constant in Orlicz Sequence Spaces Equipped with the p-Amemiya Norm

The problem of packing spheres in Orlicz sequence spacelΦ,pequipped with the p-Amemiya norm is studied, and a geometric characteristic about the reflexivity oflΦ,pis obtained, which contains the relevant work aboutlp (p&gt;1)and classical Orlicz spaceslΦdiscussed by Rankin, Burlak, and Cleaver. Moreover the packing constant as well as Kottman constant in this kind of spaces is calculated.

K-extreme point of generalized orlicz sequence spaces with Luxemburg norm

In this paper,we give necessary and sufficient conditions in order that a point \(u\in S(l_{({\it \Phi})})\) is a k-extreme point in generalized Orlicz sequence spaces equipped with the Luxemburg norm, combing the methods used in classical Orlicz spaces and new methods introduced especially for generalized ones. The results indicate the difference between the classical Orlicz spaces and generalized Orlicz spaces.

Orlicz sequence spaces spanned by identically distributed independent random variables in [formula omitted]-spaces

Let 1⩽p<2 and let Lp=Lp[0,1] be the classical Lp-space of all (classes of) p-integrable functions on [0,1]. It is known that any subspace in Lp spanned by a sequence of independent copies of a mean zero random variable f∈Lp is isomorphic to some Orlicz sequence space lN. We employ methods from interpolation theory to fully describe the class of Orlicz functions N which can be obtained by this procedure and to identify the precise connection between N and the distribution of f. Our methods strengthen and complement classical results of M.I. Kadec as well as those of J. Bretagnolle and D. Dacunha-Castelle.

Lacunary Fourier and Walsh–Fourier series near $$L^1$$ L 1

We prove that, for functions in the Orlicz class LloglogLloglogloglogL, lacunary subsequences of the Fourier and the Walsh-Fourier series converge almost everywhere. Our integrability condition is less stringent than the homologous assumption in the almost everywhere convergence theorems of Lie (Fourier case) and Do-Lacey (Walsh-Fourier case), where the quadruple logarithmic term is replaced by a triple logarithm. Our proof of the Walsh-Fourier case is self-contained and, in antithesis to Do and Lacey's argument, avoids the use of Antonov's lemma, arguing directly via novel weak-L^p bounds for the Walsh-Carleson operator.

New formulas for decreasing rearrangements and a class of Orlicz–Lorentz spaces

Using a nonlinear version of the well known Hardy–Littlewood inequalities, we derive new formulas for decreasing rearrangements of functions and sequences in the context of convex functions. We use these formulas for deducing several properties of the modular functionals defining the function and sequence spaces \(M_{\varphi ,w}\) and \(m_{\varphi ,w}\) respectively, introduced earlier in Hudzik et al. (Proc Am Math Soc 130(6): 1645–1654, 2002) for describing the Kothe dual of ordinary Orlicz–Lorentz spaces in a large variety of cases (\(\varphi \) is an Orlicz function and \(w\) a decreasing weight). We study these \(M_{\varphi ,w}\) classes in the most general setting, where they may even not be linear, and identify their Kothe duals with ordinary (Banach) Orlicz–Lorentz spaces. We introduce a new class of rearrangement invariant Banach spaces \(\mathcal M _{\varphi ,w}\) which proves to be the Kothe biduals of the \(M_{\varphi ,w}\) classes. In the case when the class \(M_{\varphi ,w}\) is a separable quasi-Banach space, \(\mathcal M _{\varphi ,w}\) is its Banach envelope.

On some geometric properties of generalized Orlicz–Lorentz sequence spaces

In this paper, we continue investigations concerning generalized Orlicz–Lorentz sequence spaces λφ initiated in the papers of Foralewski et al. (2008) [16,17] (cf. also Foralewski (2011) [11,12]). As we will show in Examples 1.1–1.3 the class of generalized Orlicz–Lorentz sequence spaces is much more wider than the class of classical Orlicz–Lorentz sequence spaces. Moreover, it is shown that if a Musielak–Orlicz function φ satisfies condition δ2λ, then λφ has the coordinatewise Kadec–Klee property. Next, monotonicity properties are considered. In order to get sufficient conditions for uniform monotonicity of the space λφ, a strong condition of δ2 type and the notion of regularity of function φ are introduced. Finally, criteria for non-squareness of λφ, of their subspaces of order continuous elements (λφ)a as well as of finite dimensional subspaces λφn of λφ are presented.

Non-squareness properties of Orlicz–Lorentz sequence spaces

In this paper criteria for non-squareness properties (non-squareness, local uniform non-squareness and uniform non-squareness) of Orlicz–Lorentz sequence spaces λφ,ω and of their n-dimensional subspaces λφ,ωn (n⩾2) as well as of the subspaces (λφ,ω)a of all order continuous elements in λφ,ω are given. Since degenerate Orlicz functions φ and degenerate weight sequences ω are also admitted, these investigations concern the most possible wide class of Orlicz–Lorentz sequence spaces. Finally, as immediate consequences, criteria for all non-squareness properties of Orlicz sequence spaces, which complete the results of Sundaresan (1966) [53], Hudzik (1985) [23], Hudzik (1985) [24], are deduced. It is worth recalling that uniform non-squareness is an important property, because it implies super-reflexivity as well as the fixed point property (see James (1964) [31], James (1972) [33] and García-Falset et al. (2006) [19]).

On approximation in weighted Smirnov–Orlicz classes

In this work, we investigate the approximation problems in weighted Smirnov–Orlicz classes. We prove a direct theorem for polynomial approximation of functions in certain subclasses of weighted Smirnov–Orlicz classes. The direct theorem is proved in terms of the modulus of smoothness. Also, from the main theorem some results are obtained. In the proof of the main theorem, a Jackson-Dzjadyk polynomial is used.

Renormalized solutions of nonlinear elliptic problems in generalized Orlicz spaces

We study a general class of nonlinear elliptic problems associated with the differential inclusion β(x,u)−div(a(x,∇u)+F(u))∋f, where f∈L1(Ω). The vector field a(⋅,⋅) is monotone in the second variable and satisfies a non-standard growth condition described by an x-dependent convex function that generalizes both Lp(x) and classical Orlicz settings. Using truncation techniques and a generalized Minty method in the functional setting of non-reflexive spaces we prove existence of renormalized solutions for general L1-data. Under an additional strict monotonicity assumption uniqueness of the renormalized solution is established. Sufficient conditions are specified which guarantee that the renormalized solution is already a weak solution to the problem.

Explosiveness of Musielak-Orlicz Sequence Spaces

In this talk,we make a survey for criteria of the exposed points and the exposedness for the sequence spaces of Orlicz type,such as the classical Orlicz spaces,general Orlicz spaces and Musielak-Orlicz spaces,endowed with Luxemburg norm and Orlicz norm respectively.It is well known that explosiveness is basic concept in the geometric theory of Banach spaces.They have numerous application in separation theory and control theory.Criteria for exposed points and strongly exposed points in all classical Orlicz spaces were given in[10-12].In recent years,Zhao and Cui in[6]discussed the problem in Musielak-Orlicz sequence spaces under some restriction. By counterexamples,we show that the criteria of extreme points in[8]and exposed points in[6] are not true,and we give the criteria for exposed points and strongly exposed points in arbitrary Musielak-Orlicz sequence spaces equipped with Luxemburg norm and Orlicz norm by getting rid of the restriction on Musielak-Orlicz function in[6].