Displaying similar documents to “Calderón couples of rearrangement invariant spaces”

Fenchel-Orlicz spaces

Barry Turett

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

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 , ) with Orlicz norm, and get the following inequality: B Φ B Φ - 1 q Φ Q Φ 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 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 I φ ( ( f * ) / w ) w ,where (f*)⁰ is Halperin’s level...

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 < ∞.

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 ) W ¹ L M ( Ω ) , μ L ¹ ( Ω ) + W - 1 E M ̅ ( Ω ) and ϕ C ( , 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 .

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 Φ ) * will be presented.

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.

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 ) L φ ( G ) × L ψ ( G ) : f * g is well defined on G is σ-c-lower porous in L φ ( G ) × 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 , ) (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, v ϕ ( u ) · w < 2 a n d u < ) . (2) Λ ϕ , w has weakly normal structure if and only if 0 v ϕ ( u ) · w < 2 .

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...

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 | | ( x i X i ) i = 1 | | p 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 p -norm by an arbitrary...

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 ) ) ( 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.

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.

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 N , N ≥ 3, is a smooth bounded domain, ν is the outward unit normal to ∂Ω, ρ ( u ) = Ω ( Φ ( | u | ) + Φ ( | u | ) ) d x , M: [0,∞) → ℝ is a continuous function, K L ( 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.

Gagliardo-Nirenberg inequalities in logarithmic spaces

Agnieszka Kałamajska, Katarzyna Pietruska-Pałuba (2006)

Colloquium Mathematicae

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We obtain interpolation inequalities for derivatives: M q , α ( | f ( x ) | ) d x C [ M p , β ( Φ ( x , | f | , | ( 2 ) f | ) ) d x + M r , γ ( Φ ( x , | f | , | ( 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 | | · | | ( 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...

Khinchin inequality and Banach-Saks type properties in rearrangement-invariant spaces

F. A. Sukochev, D. Zanin (2009)

Studia Mathematica

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We study the class of all rearrangement-invariant ( = r.i.) function spaces E on [0,1] such that there exists 0 < q < 1 for which k = 1 n ξ k E C n q , where ξ k k 1 E is an arbitrary sequence of independent identically distributed symmetric random variables on [0,1] and C > 0 does not depend on n. We completely characterize all Lorentz spaces having this property and complement classical results of Rodin and Semenov for Orlicz spaces e x p ( L p ) , p ≥ 1. We further apply our results to the study of Banach-Saks...

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...

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 M ( M ) are uniformly isomorphic to subspaces of L₁ if M₁ and M₂ are “separated” by a function t r , 1 < r < 2.

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 ( ) 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 ( ) 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 ( ) , and no Orlicz...

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

Jakub Onufry Wojtaszczyk (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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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 , . . . , X i k ) g ( X j , . . . , X j l ) f ( X i , . . . , X i k ) g ( X j , . . . , 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...

Explicit formulas for optimal rearrangement-invariant norms in Sobolev imbedding inequalities

Ron Kerman, Luboš Pick (2011)

Studia Mathematica

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We study imbeddings of the Sobolev space W m , ϱ ( Ω ) : = u: Ω → ℝ with ϱ ( α u / x α ) < ∞ when |α| ≤ m, in which Ω is a bounded Lipschitz domain in ℝⁿ, ϱ is a rearrangement-invariant (r.i.) norm and 1 ≤ m ≤ n - 1. For such a space we have shown there exist r.i. norms, τ ϱ and σ ϱ , that are optimal with respect to the inclusions W m , ϱ ( Ω ) W m , τ ϱ ( Ω ) L σ ϱ ( Ω ) . General formulas for τ ϱ and σ ϱ are obtained using the -method of interpolation. These lead to explicit expressions when ϱ is a Lorentz Gamma norm or an Orlicz norm.

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 , | | · | | X ) . Recall that a linear operator T : L Φ X is said to be σ-smooth whenever u ( o ) 0 in L Φ implies | | T ( u ) | | X 0 . It is shown that every σ-smooth operator T : L Φ X factors through the inclusion map j : L Φ L Φ ̅ , where Φ̅ denotes the convex minorant of Φ. We obtain the Bochner integral representation of σ-smooth operators T : L Φ X . This extends some earlier results of J. J. Uhl concerning the Bochner integral representation of linear operators defined on...

Some results on packing in Orlicz sequence spaces

Y. Q. Yan (2001)

Studia Mathematica

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We present monotonicity theorems for index functions of N-fuctions, and obtain formulas for exact values of packing constants. In particular, we show that the Orlicz sequence space l ( N ) generated by the N-function N(v) = (1+|v|)ln(1+|v|) - |v| with Luxemburg norm has the Kottman constant K ( l ( N ) ) = N - 1 ( 1 ) / N - 1 ( 1 / 2 ) , which answers M. M. Rao and Z. D. Ren’s [8] problem.

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 H ( ) : ψ ( | f ( z ) | ) d A α ( z ) < , 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 ψ ( t ) / t = 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 .