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

Displaying similar documents to “The canonical injection of the Hardy-Orlicz space $H^{Ψ}$ into the Bergman-Orlicz space $^{Ψ}$ ”

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

Nonlinear unilateral problems in Orlicz spaces

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

Applicationes Mathematicae

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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 $ϕ \in C ⁰ (ℝ, ℝ^{N})$ .

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 $S_{F}^{φ}$ .

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...

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.

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 < ∞.

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.

Calderón couples of rearrangement invariant spaces

N. Kalton (1993)

Studia Mathematica

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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...

Geometry of Orlicz spaces

Chen Shutao

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

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.

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

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.

A Hardy space related to the square root of the Poisson kernel

Jonatan Vasilis (2010)

Studia Mathematica

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A real-valued Hardy space $H ¹_{\sqrt} () \subseteq L ¹ ()$ related to the square root of the Poisson kernel in the unit disc is defined. The space is shown to be strictly larger than its classical counterpart H¹(). A decreasing function is in $H ¹_{\sqrt} ()$ if and only if the function is in the Orlicz space LloglogL(). In contrast to the case of H¹(), there is no such characterization for general positive functions: every Orlicz space strictly larger than L log L() contains positive functions which do not belong to $H ¹_{\sqrt} ()$ , and no Orlicz...

Complex Convexity of Orlicz-Lorentz Spaces and its Applications

Changsun Choi, Anna Kamińska, Han Ju Lee (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

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We give sufficient and necessary conditions for complex extreme points of the unit ball of Orlicz-Lorentz spaces, as well as we find criteria for the complex rotundity and uniform complex rotundity of these spaces. As an application we show that the set of norm-attaining operators is dense in the space of bounded linear operators from $d *_{(} w, 1)$ into d(w,1), where $d *_{(} w, 1)$ is a predual of a complex Lorentz sequence space d(w,1), if and only if wi ∈ c₀∖ℓ₂.

Compactness of composition operators acting on weighted Bergman-Orlicz spaces

Ajay K. Sharma, S. Ueki (2012)

Annales Polonici Mathematici

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We characterize compact composition operators acting on weighted Bergman-Orlicz spaces $_{α}^{ψ} = f \in H () : \int_{} ψ (| f (z) |) d A_{α} (z) < \infty$ , where α > -1 and ψ is a strictly increasing, subadditive convex function defined on [0,∞) and satisfying ψ(0) = 0, the growth condition $l i m_{t \to \infty} ψ (t) / t = \infty$ and the Δ₂-condition. In fact, we prove that $C_{φ}$ is compact on $_{α}^{ψ}$ if and only if it is compact on the weighted Bergman space $²_{α}$ .

On the distribution of random variables corresponding to Musielak-Orlicz norms

David Alonso-Gutiérrez, Sören Christensen, Markus Passenbrunner, Joscha Prochno (2013)

Studia Mathematica

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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 X₁,...,Xₙ are independent copies of X, then $1 / C_{p} {| | x | |}_{M} \leq | | (x_{i} X_{i}) ⁿ_{i = 1} {| |}_{p} \leq C_{p} {| | x | |}_{M}$ , where $C_{p}$ is a positive constant depending only on p. In case p = 2 we need the function t ↦ tM’(t) - M(t) to be 2-concave and as an application immediately obtain an embedding of the corresponding Orlicz spaces into L₁[0,1]. We also provide a general result replacing the $ℓ_{p}$ -norm by an arbitrary...

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.

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

Ryszard Grząślewicz (1987)

Colloquium Mathematicae

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Hankel operators and weak factorization for Hardy-Orlicz spaces

Aline Bonami, Sandrine Grellier (2010)

Colloquium Mathematicae

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We study the holomorphic Hardy-Orlicz spaces $^{Φ} (Ω)$ , where Ω is the unit ball or, more generally, a convex domain of finite type or a strictly pseudoconvex domain in ℂⁿ. The function Φ is in particular such that $¹ (Ω) \subset^{Φ} (Ω) \subset^{p} (Ω)$ for some p > 0. We develop maximal characterizations, atomic and molecular decompositions. We then prove weak factorization theorems involving the space BMOA(Ω). As a consequence, we characterize those Hankel operators which are bounded from $^{Φ} (Ω)$ into ¹(Ω).

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

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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 $Φ_{0}$ .

Gagliardo-Nirenberg inequalities in logarithmic spaces

Agnieszka Kałamajska, Katarzyna Pietruska-Pałuba (2006)

Colloquium Mathematicae

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We obtain interpolation inequalities for derivatives: $\int M_{q, α} (| \nabla f (x) |) d x \leq C [\int M_{p, β} (Φ ₁ (x, | f |, | \nabla^{(2)} f |)) d x + \int M_{r, γ} (Φ ₂ (x, | f |, | \nabla^{(2)} f |)) d x]$ , and their counterparts expressed in Orlicz norms: ||∇f||²(q,α) ≤ C||Φ₁(x,|f|,|∇(2)f|)||(p,β) ||Φ₂(x,|f|,|∇(2)f|)||(r,γ) $,$ where ${| | \cdot | |}_{(s, κ)}$ is the Orlicz norm relative to the function $M_{s, κ} (t) = t^{s} {(l n (2 + t))}^{κ}$ . The parameters p,q,r,α,β,γ and the Carathéodory functions Φ₁,Φ₂ are supposed to satisfy certain consistency conditions. Some of the classical Gagliardo-Nirenberg inequalities follow as a special case. Gagliardo-Nirenberg inequalities in logarithmic spaces...

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.

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...

Compactness of Sobolev imbeddings involving rearrangement-invariant norms

Ron Kerman, Luboš Pick (2008)

Studia Mathematica

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We find necessary and sufficient conditions on a pair of rearrangement-invariant norms, ϱ and σ, in order that the Sobolev space $W^{m, ϱ} (Ω)$ be compactly imbedded into the rearrangement-invariant space $L_{σ} (Ω)$ , where Ω is a bounded domain in ℝⁿ with Lipschitz boundary and 1 ≤ m ≤ n-1. In particular, we establish the equivalence of the compactness of the Sobolev imbedding with the compactness of a certain Hardy operator from $L_{ϱ} (0, | Ω |)$ into $L_{σ} (0, | Ω |)$ . The results are illustrated with examples in which ϱ and σ are both...

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...

Combinatorial inequalities and subspaces of L₁

Joscha Prochno, Carsten Schütt (2012)

Studia Mathematica

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Let M₁ and M₂ be N-functions. We establish some combinatorial inequalities and show that the product spaces $ℓ ⁿ_{M ₁} (ℓ ⁿ_{M ₂})$ are uniformly isomorphic to subspaces of L₁ if M₁ and M₂ are “separated” by a function $t^{r}$ , 1 < r < 2.

A Simpler Proof of the Negative Association Property for Absolute Values of Measures Tied to Generalized Orlicz Balls

Jakub Onufry Wojtaszczyk (2009)

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Negative association for a family of random variables $(X_{i})$ means that for any coordinatewise increasing functions f,g we have $(X_{i ₁}, . . ., X_{i_{k}}) g (X_{j ₁}, . . ., X_{j_{l}}) \leq f (X_{i ₁}, . . ., X_{i_{k}}) g (X_{j ₁}, . . ., X_{j_{l}})$ for any disjoint sets of indices (iₘ), (jₙ). It is a way to indicate the negative correlation in a family of random variables. It was first introduced in 1980s in statistics by Alem Saxena and Joag-Dev Proschan, and brought to convex geometry in 2005 by Wojtaszczyk Pilipczuk to prove the Central Limit Theorem for Orlicz balls. The paper gives a relatively simple...

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Zhongrui Shi, Chunyan Liu (2011)

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Some results on packing in Orlicz sequence spaces

Y. Q. Yan (2001)

Studia Mathematica

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

Marek Nawrocki (1992)

Studia Mathematica

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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 $L N_{*} (^{n})$ .

Displaying similar documents to “The canonical injection of the Hardy-Orlicz space H Ψ into the Bergman-Orlicz space Ψ ”

Displaying similar documents to “The canonical injection of the Hardy-Orlicz space $H^{Ψ}$ into the Bergman-Orlicz space $^{Ψ}$ ”