Weak difference properties of higher orders for the class $L_p(G)$

Zbigniew Gajda

Displaying similar documents to “Weak difference properties of higher orders for the class $L_{p} (G)$ ”

On certain porous sets in the Orlicz space of a locally compact group

Ibrahim Akbarbaglu, Saeid Maghsoudi (2012)

Colloquium Mathematicae

Similarity:

Let G be a locally compact group with a fixed left Haar measure. Given Young functions φ and ψ, we consider the Orlicz spaces $L^{φ} (G)$ and $L^{ψ} (G)$ on a non-unimodular group G, and, among other things, we prove that under mild conditions on φ and ψ, the set $(f, g) \in L^{φ} (G) \times L^{ψ} (G) : f * g$ is well defined on G is σ-c-lower porous in $L^{φ} (G) \times L^{ψ} (G)$ . This answers a question raised by Głąb and Strobin in 2010 in a more general setting of Orlicz spaces. We also prove a similar result for non-compact locally compact groups.

Weakly compact sets in Orlicz sequence spaces

Siyu Shi, Zhong Rui Shi, Shujun Wu (2021)

Czechoslovak Mathematical Journal

Similarity:

We combine the techniques of sequence spaces and general Orlicz functions that are broader than the classical cases of $N$ -functions. We give three criteria for the weakly compact sets in general Orlicz sequence spaces. One criterion is related to elements of dual spaces. Under the restriction of ${lim}_{u \to 0} M (u) / u = 0$ , we propose two other modular types that are convenient to use because they get rid of elements of dual spaces. Subsequently, by one of these two modular criteria, we see that a set $A$ in Riesz...

Decomposable sets and Musielak-Orlicz spaces of multifunctions

Andrzej Kasperski (2005)

Banach Center Publications

Similarity:

We introduce the Musielak-Orlicz space of multifunctions $X_{m, φ}$ and the set $S_{F}^{φ}$ of φ-integrable selections of F. We show that some decomposable sets in Musielak-Orlicz space belong to $X_{m, φ}$ . We generalize Theorem 3.1 from [6]. Also, we get some theorems on the space $X_{m, φ}$ and the set $S_{F}^{φ}$ .

Nonlinear unilateral problems in Orlicz spaces

L. Aharouch, E. Azroul, M. Rhoudaf (2006)

Applicationes Mathematicae

Similarity:

We prove the existence of solutions of the unilateral problem for equations of the type Au - divϕ(u) = μ in Orlicz spaces, where A is a Leray-Lions operator defined on $(A) \subset W ₀ ¹ L_{M} (Ω)$ , $μ \in L ¹ (Ω) + W^{- 1} E_{M ̅} (Ω)$ and $ϕ \in C ⁰ (ℝ, ℝ^{N})$ .

An inequality in Orlicz function spaces with Orlicz norm

Jin Cai Wang (2003)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

We use Simonenko quantitative indices of an $𝒩$ -function $Φ$ to estimate two parameters $q_{Φ}$ and $Q_{Φ}$ in Orlicz function spaces $L^{Φ} [0, \infty)$ with Orlicz norm, and get the following inequality: $\frac{B_{Φ}}{B_{Φ} - 1} \leq q_{Φ} \leq Q_{Φ} \leq \frac{A_{Φ}}{A_{φ} - 1}$ , where $A_{Φ}$ and $B_{Φ}$ are Simonenko indices. A similar inequality is obtained in $L^{Φ} [0, 1]$ with Orlicz norm.

Normal structure of Lorentz-Orlicz spaces

Pei-Kee Lin, Huiying Sun (1997)

Annales Polonici Mathematici

Similarity:

Let ϕ: ℝ → ℝ₊ ∪ 0 be an even convex continuous function with ϕ(0) = 0 and ϕ(u) > 0 for all u > 0 and let w be a weight function. u₀ and v₀ are defined by u₀ = supu: ϕ is linear on (0,u), v₀=supv: w is constant on (0,v) (where sup∅ = 0). We prove the following theorem. Theorem. Suppose that $Λ_{ϕ, w} (0, \infty)$ (respectively, $Λ_{ϕ, w} (0, 1)$ ) is an order continuous Lorentz-Orlicz space. (1) $Λ_{ϕ, w}$ has normal structure if and only if u₀ = 0 (respectively, $\int^{v ₀} ϕ (u ₀) \cdot w < 2 a n d u ₀ < \infty) .$ (2) $Λ_{ϕ, w}$ has weakly normal structure if and only if $\int_{0}^{v ₀} ϕ (u ₀) \cdot w < 2$ .

Lower bounds for Jung constants of Orlicz sequence spaces

Z. D. Ren (2010)

Annales Polonici Mathematici

Similarity:

A new lower bound for the Jung constant $J C (l^{(Φ)})$ of the Orlicz sequence space $l^{(Φ)}$ defined by an N-function Φ is found. It is proved that if $l^{(Φ)}$ is reflexive and the function tΦ’(t)/Φ(t) is increasing on $(0, Φ^{- 1} (1)]$ , then $J C (l^{(Φ)}) \geq (Φ^{- 1} (1 / 2)) / (Φ^{- 1} (1))$ . Examples in Section 3 show that the above estimate is better than in Zhang’s paper (2003) in some cases and that the results given in Yan’s paper (2004) are not accurate.

Dual spaces to Orlicz-Lorentz spaces

Anna Kamińska, Karol Leśnik, Yves Raynaud (2014)

Studia Mathematica

Similarity:

For an Orlicz function φ and a decreasing weight w, two intrinsic exact descriptions are presented for the norm in the Köthe dual of the Orlicz-Lorentz function space $Λ_{φ, w}$ or the sequence space $λ_{φ, w}$ , equipped with either the Luxemburg or Amemiya norms. The first description is via the modular $i n f \int φ ⁎ (f * / | g |) | g | : g ≺ w$ , where f* is the decreasing rearrangement of f, ≺ denotes submajorization, and φ⁎ is the complementary function to φ. The second description is in terms of the modular $\int_{I} φ ⁎ ((f *) ⁰ / w) w$ ,where (f*)⁰ is Halperin’s level...

Generalized gradients for locally Lipschitz integral functionals on non- $L^{p}$ -type spaces of measurable functions

Hôǹg Thái Nguyêñ, Dariusz Pączka (2008)

Banach Center Publications

Similarity:

Let (Ω,μ) be a measure space, E be an arbitrary separable Banach space, $E *_{ω *}$ be the dual equipped with the weak* topology, and g:Ω × E → ℝ be a Carathéodory function which is Lipschitz continuous on each ball of E for almost all s ∈ Ω. Put $G (x) : = \int_{Ω} g (s, x (s)) d μ (s)$ . Consider the integral functional G defined on some non- $L^{p}$ -type Banach space X of measurable functions x: Ω → E. We present several general theorems on sufficient conditions under which any element γ ∈ X* of Clarke’s generalized gradient (multivalued...

Linear operators on non-locally convex Orlicz spaces

Marian Nowak, Agnieszka Oelke (2008)

Banach Center Publications

Similarity:

We study linear operators from a non-locally convex Orlicz space $L^{Φ}$ to a Banach space $(X, | | \cdot | |_{X})$ . Recall that a linear operator $T : L^{Φ} \to X$ is said to be σ-smooth whenever $u ₙ ⟶^{(o)} 0$ in $L^{Φ}$ implies ${| | T (u ₙ) | |}_{X} \to 0$ . It is shown that every σ-smooth operator $T : L^{Φ} \to X$ factors through the inclusion map $j : L^{Φ} \to L^{Φ ̅}$ , where Φ̅ denotes the convex minorant of Φ. We obtain the Bochner integral representation of σ-smooth operators $T : L^{Φ} \to X$ . This extends some earlier results of J. J. Uhl concerning the Bochner integral representation of linear operators defined on...

Fenchel-Orlicz spaces

Barry Turett

Similarity:

CONTENTSIntroduction............................................................................... 51. Definitions and preliminary results......................................... 72. Completeness of $L^{Φ} (μ, ℜ)$ .............................. 93. Linear functionals on $L^{Φ} (μ, ℜ)$ ....................... 264. Geometry of Fenchel-Orlicz spaces........................................ 41References....................................................................................... 54

The space of multipliers and convolutors of Orlicz spaces on a locally compact group

Hasan P. Aghababa, Ibrahim Akbarbaglu, Saeid Maghsoudi (2013)

Studia Mathematica

Similarity:

Let G be a locally compact group, let (φ,ψ) be a complementary pair of Young functions, and let $L^{φ} (G)$ and $L^{ψ} (G)$ be the corresponding Orlicz spaces. Under some conditions on φ, we will show that for a Banach $L^{φ} (G)$ -submodule X of $L^{ψ} (G)$ , the multiplier space $H o m_{L^{φ} (G)} (L^{φ} (G), X *)$ is a dual Banach space with predual $L^{φ} (G) ∙ X : = \bar{s p a n} u x : u \in L^{φ} (G), x \in X$ , where the closure is taken in the dual space of $H o m_{L^{φ} (G)} (L^{φ} (G), X *)$ . We also prove that if $φ$ is a Δ₂-regular N-function, then $C v_{φ} (G)$ , the space of convolutors of $M^{φ} (G)$ , is identified with the dual of a Banach algebra of functions on G...

Uniform convexity and associate spaces

Petteri Harjulehto, Peter Hästö (2018)

Czechoslovak Mathematical Journal

Similarity:

We prove that the associate space of a generalized Orlicz space $L^{φ (\cdot)}$ is given by the conjugate modular $φ^{*}$ even without the assumption that simple functions belong to the space. Second, we show that every weakly doubling $Φ$ -function is equivalent to a doubling $Φ$ -function. As a consequence, we conclude that $L^{φ (\cdot)}$ is uniformly convex if $φ$ and $φ^{*}$ are weakly doubling.

Trudinger's inequality for double phase functionals with variable exponents

Fumi-Yuki Maeda, Yoshihiro Mizuta, Takao Ohno, Tetsu Shimomura (2021)

Czechoslovak Mathematical Journal

Similarity:

Our aim in this paper is to establish Trudinger’s inequality on Musielak-Orlicz-Morrey spaces $L^{Φ, κ} (G)$ under conditions on $Φ$ which are essentially weaker than those considered in a former paper. As an application and example, we show Trudinger’s inequality for double phase functionals $Φ (x, t) = t^{p (x)} + a (x) t^{q (x)}$ , where $p (\cdot)$ and $q (\cdot)$ satisfy log-Hölder conditions and $a (\cdot)$ is nonnegative, bounded and Hölder continuous.

The canonical injection of the Hardy-Orlicz space $H^{Ψ}$ into the Bergman-Orlicz space $^{Ψ}$

Pascal Lefèvre, Daniel Li, Hervé Queffélec, Luis Rodríguez-Piazza (2011)

Studia Mathematica

Similarity:

We study the canonical injection from the Hardy-Orlicz space $H^{Ψ}$ into the Bergman-Orlicz space $^{Ψ}$ .

Generalization of the weak amenability on various Banach algebras

Madjid Eshaghi Gordji, Ali Jabbari, Abasalt Bodaghi (2019)

Mathematica Bohemica

Similarity:

The generalized notion of weak amenability, namely $(ϕ, ψ)$ -weak amenability, where $ϕ, ψ$ are continuous homomorphisms on a Banach algebra $𝒜$ , was introduced by Bodaghi, Eshaghi Gordji and Medghalchi (2009). In this paper, the $(ϕ, ψ)$ -weak amenability on the measure algebra $M (G)$ , the group algebra $L^{1} (G)$ and the Segal algebra $S^{1} (G)$ , where $G$ is a locally compact group, are studied. As a typical example, the $(ϕ, ψ)$ -weak amenability of a special semigroup algebra is shown as well.

Linear topological properties of the Lumer-Smirnov class of the polydisc

Marek Nawrocki (1992)

Studia Mathematica

Similarity:

Linear topological properties of the Lumer-Smirnov class $L N_{*} (^{n})$ of the unit polydisc $^{n}$ are studied. The topological dual and the Fréchet envelope are described. It is proved that $L N_{*} (^{n})$ has a weak basis but it is nonseparable in its original topology. Moreover, it is shown that the Orlicz-Pettis theorem fails for $L N_{*} (^{n})$ .

Existence of solutions for a class of Kirchhoff type problems in Orlicz-Sobolev spaces

Nguyen Thanh Chung (2015)

Annales Polonici Mathematici

Similarity:

We consider Kirchhoff type problems of the form ⎧ -M(ρ(u))(div(a(|∇u|)∇u) - a(|u|)u) = K(x)f(u) in Ω ⎨ ⎩ ∂u/∂ν = 0 on ∂Ω where $Ω \subset ℝ^{N}$ , N ≥ 3, is a smooth bounded domain, ν is the outward unit normal to ∂Ω, $ρ (u) = \int_{Ω} (Φ (| \nabla u |) + Φ (| u |)) d x$ , M: [0,∞) → ℝ is a continuous function, $K \in L^{\infty} (Ω)$ , and f: ℝ → ℝ is a continuous function not satisfying the Ambrosetti-Rabinowitz type condition. Using variational methods, we obtain some existence and multiplicity results.

Periodic solutions of Euler-Lagrange equations with sublinear potentials in an Orlicz-Sobolev space setting

Sonia Acinas, Fernando Mazzone (2017)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

Similarity:

In this paper, we obtain existence results of periodic solutions of hamiltonian systems in the Orlicz-Sobolev space $W^{1} L^{Φ} ([0, T])$ . We employ the direct method of calculus of variations and we consider a potential function $F$ satisfying the inequality $| \nabla F (t, x) | \leq b_{1} (t) Φ_{0}^{'} (| x |) + b_{2} (t)$ , with $b_{1}, b_{2} \in L^{1}$ and certain $N$ -functions $Φ_{0}$ .

Displaying similar documents to “Weak difference properties of higher orders for the class L p ( G ) ”

Displaying similar documents to “Weak difference properties of higher orders for the class $L_{p} (G)$ ”