Hankel operators and weak factorization for Hardy-Orlicz spaces

Aline Bonami; Sandrine Grellier

Displaying similar documents to “Hankel operators and weak factorization for Hardy-Orlicz spaces”

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 $^{Ψ}$ .

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

Ryszard Grząślewicz (1987)

Colloquium Mathematicae

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

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.

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...

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 $ϕ \in C ⁰ (ℝ, ℝ^{N})$ .

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 $S_{F}^{φ}$ .

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...

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...

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.

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.

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.

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...

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

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.

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.

Trudinger's inequality for double phase functionals with variable exponents

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

Czechoslovak Mathematical Journal

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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.

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...

Toeplitz operators on Bergman spaces and Hardy multipliers

Wolfgang Lusky, Jari Taskinen (2011)

Studia Mathematica

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We study Toeplitz operators $T_{a}$ with radial symbols in weighted Bergman spaces $A_{μ}^{p}$ , 1 < p < ∞, on the disc. Using a decomposition of $A_{μ}^{p}$ into finite-dimensional subspaces the operator $T_{a}$ can be considered as a coefficient multiplier. This leads to new results on boundedness of $T_{a}$ and also shows a connection with Hardy space multipliers. Using another method we also prove a necessary and sufficient condition for the boundedness of $T_{a}$ for a satisfying an assumption on the positivity of certain...