Complex Convexity of Orlicz-Lorentz Spaces and its Applications

Changsun Choi; Anna Kamińska; Han Ju Lee

Displaying similar documents to “Complex Convexity of Orlicz-Lorentz Spaces and its Applications”

On isometric domains of positive operators on $L^{p}$ -spaces

Ryszard Grząślewicz (1987)

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An inequality in Orlicz function spaces with Orlicz norm

Jin Cai Wang (2003)

Commentationes Mathematicae Universitatis Carolinae

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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.

Decomposable sets and Musielak-Orlicz spaces of multifunctions

Andrzej Kasperski (2005)

Banach Center Publications

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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 $S_{F}^{φ}$ .

On some global and local geometric properties of Calderón-Lozanovskiĭ spaces

Agata Narloch (2005)

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Criteria for full k-rotundity (k ∈ ℕ, k ≥ 2) and uniform rotundity in every direction of Calderón-Lozanovskiĭ spaces are formulated. A characterization of $H_{μ}$ -points in these spaces is also given.

Nonlinear unilateral problems in Orlicz spaces

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

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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 $ϕ \in C ⁰ (ℝ, ℝ^{N})$ .

Fenchel-Orlicz spaces

Barry Turett

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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

Dual spaces to Orlicz-Lorentz spaces

Anna Kamińska, Karol Leśnik, Yves Raynaud (2014)

Studia Mathematica

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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...

Geometry of Orlicz spaces

Chen Shutao

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CONTENTSPreface..............................................................................................................................4Introduction........................................................................................................................51. Orlicz spaces..................................................................................................................6 1.1. Orlicz functions...........................................................................................................6 1.2....

Normal structure of Lorentz-Orlicz spaces

Pei-Kee Lin, Huiying Sun (1997)

Annales Polonici Mathematici

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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 $\int_{0}^{v ₀} ϕ (u ₀) \cdot w < 2$ .

On some properties for dual spaces of Musielak-Orlicz function spaces

Zenon Zbąszyniak (2011)

Banach Center Publications

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We will present relationships between the modular ρ* and the norm in the dual spaces $(L_{Φ}) *$ in the case when a Musielak-Orlicz space $L_{Φ}$ is equipped with the Orlicz norm. Moreover, criteria for extreme points of the unit sphere of the dual space $(L ⁰_{Φ}) *$ will be presented.

An operator characterization of $L^{p}$ -spaces in a class of Orlicz spaces

Maciej Burnecki (2008)

Banach Center Publications

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We consider an embedding of the group of invertible transformations of [0,1] into the algebra of bounded linear operators on an Orlicz space. We show that if this embedding preserves the group action then the Orlicz space is an $L^{p}$ -space for some 1 ≤ p < ∞.

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

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

A remark on (p, q)-absolutely summing operators in $l^{p}$ -spaces

Ron C. Blei (1982)

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Lower bounds for Jung constants of Orlicz sequence spaces

Z. D. Ren (2010)

Annales Polonici Mathematici

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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.

Linear operators on non-locally convex Orlicz spaces

Marian Nowak, Agnieszka Oelke (2008)

Banach Center Publications

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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...

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

Ibrahim Akbarbaglu, Saeid Maghsoudi (2012)

Colloquium Mathematicae

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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.

Uniform convexity and associate spaces

Petteri Harjulehto, Peter Hästö (2018)

Czechoslovak Mathematical Journal

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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.

Reflexive subspaces of some Orlicz spaces

Emmanuelle Lavergne (2008)

Colloquium Mathematicae

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We show that when the conjugate of an Orlicz function ϕ satisfies the growth condition Δ⁰, then the reflexive subspaces of $L^{ϕ}$ are closed in the L¹-norm. For that purpose, we use (and give a new proof of) a result of J. Alexopoulos saying that weakly compact subsets of such $L^{ϕ}$ have equi-absolutely continuous norm.

Weakly compact sets in Orlicz sequence spaces

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

Czechoslovak Mathematical Journal

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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...

Calderón couples of rearrangement invariant spaces

N. Kalton (1993)

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We examine conditions under which a pair of rearrangement invariant function spaces on [0,1] or [0,∞) form a Calderón couple. A very general criterion is developed to determine whether such a pair is a Calderón couple, with numerous applications. We give, for example, a complete classification of those spaces X which form a Calderón couple with $L_{\infty} .$ We specialize our results to Orlicz spaces and are able to give necessary and sufficient conditions on an Orlicz function F so that the pair...

Dunford-Pettis operators on the space of Bochner integrable functions

Marian Nowak (2011)

Banach Center Publications

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Let (Ω,Σ,μ) be a finite measure space and let X be a real Banach space. Let $L^{Φ} (X)$ be the Orlicz-Bochner space defined by a Young function Φ. We study the relationships between Dunford-Pettis operators T from L¹(X) to a Banach space Y and the compactness properties of the operators T restricted to $L^{Φ} (X)$ . In particular, it is shown that if X is a reflexive Banach space, then a bounded linear operator T:L¹(X) → Y is Dunford-Pettis if and only if T restricted to $L^{\infty} (X)$ is $(τ (L^{\infty} (X), L ¹ (X *)), | | \cdot | |_{Y})$ -compact.

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

Nguyen Thanh Chung (2015)

Annales Polonici Mathematici

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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.

Orlicz boundedness for certain classical operators

E. Harboure, O. Salinas, B. Viviani (2002)

Colloquium Mathematicae

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Let ϕ and ψ be functions defined on [0,∞) taking the value zero at zero and with non-negative continuous derivative. Under very mild extra assumptions we find necessary and sufficient conditions for the fractional maximal operator $M_{Ω}^{α}$ , associated to an open bounded set Ω, to be bounded from the Orlicz space $L^{ψ} (Ω)$ into $L^{ϕ} (Ω)$ , 0 ≤ α < n. For functions ϕ of finite upper type these results can be extended to the Hilbert transform f̃ on the one-dimensional torus and to the fractional integral operator...

Boundedness of generalized fractional integral operators on Orlicz spaces near $L^{1}$ over metric measure spaces

Daiki Hashimoto, Takao Ohno, Tetsu Shimomura (2019)

Czechoslovak Mathematical Journal

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We are concerned with the boundedness of generalized fractional integral operators $I_{ρ, τ}$ from Orlicz spaces $L^{Φ} (X)$ near $L^{1} (X)$ to Orlicz spaces $L^{Ψ} (X)$ over metric measure spaces equipped with lower Ahlfors $Q$ -regular measures, where $Φ$ is a function of the form $Φ (r) = r ℓ (r)$ and $ℓ$ is of log-type. We give a generalization of paper by Mizuta et al. (2010), in the Euclidean setting. We deal with both generalized Riesz potentials and generalized logarithmic potentials.