Displaying similar documents to “Periodic solutions of Euler-Lagrange equations with sublinear potentials in an Orlicz-Sobolev space setting”

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.

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 ( · ) and q ( · ) satisfy log-Hölder conditions and a ( · ) is nonnegative, bounded and Hölder continuous.

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 .

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 ( · ) 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 ( · ) is uniformly convex if φ and φ * are weakly doubling.

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

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

Studia Mathematica

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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 : = s p a n ¯ u x : u L φ ( G ) , x 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...

Remarks on the critical Besov space and its embedding into weighted Besov-Orlicz spaces

Hidemitsu Wadade (2010)

Studia Mathematica

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We present several continuous embeddings of the critical Besov space B p n / p , ρ ( ) . We first establish a Gagliardo-Nirenberg type estimate | | u | | q , w r 0 , ν C ( 1 / ( n - r ) ) 1 / q + 1 / ν - 1 / ρ ( q / r ) 1 / ν - 1 / ρ | | u | | p 0 , ρ ( n - r ) p / n q | | u | | p n / p , ρ 1 - ( n - r ) p / n q , for 1 < p ≤ q < ∞, 1 ≤ ν < ρ ≤ ∞ and the weight function w r ( x ) = 1 / ( | x | r ) with 0 < r < n. Next, we prove the corresponding Trudinger type estimate, and obtain it in terms of the embedding B p n / p , ρ ( ) B Φ , w r 0 , ν ( ) , where the function Φ₀ of the weighted Besov-Orlicz space B Φ , w r 0 , ν ( ) is a Young function of the exponential type. Another point of interest is to embed B p n / p , ρ ( ) into the weighted Besov...

Isomorphisms and several characterizations of Musielak-Orlicz-Hardy spaces associated with some Schrödinger operators

Sibei Yang (2015)

Czechoslovak Mathematical Journal

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Let L : = - Δ + V be a Schrödinger operator on n with n 3 and V 0 satisfying Δ - 1 V L ( n ) . Assume that ϕ : n × [ 0 , ) [ 0 , ) is a function such that ϕ ( x , · ) is an Orlicz function, ϕ ( · , t ) 𝔸 ( n ) (the class of uniformly Muckenhoupt weights). Let w be an L -harmonic function on n with 0 < C 1 w C 2 , where C 1 and C 2 are positive constants. In this article, the author proves that the mapping H ϕ , L ( n ) f w f H ϕ ( n ) is an isomorphism from the Musielak-Orlicz-Hardy space associated with L , H ϕ , L ( n ) , to the Musielak-Orlicz-Hardy space H ϕ ( n ) under some assumptions on ϕ . As applications, the author further...

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.

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 ( r ) and 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.

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...

Concentration-Compactness Principle for embedding into multiple exponential spaces on unbounded domains

Robert Černý (2015)

Czechoslovak Mathematical Journal

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Let Ω n be a domain and let α < n - 1 . We prove the Concentration-Compactness Principle for the embedding of the space W 0 1 L n log α L ( Ω ) into an Orlicz space corresponding to a Young function which behaves like exp ( t n / ( n - 1 - α ) ) for large t . We also give the result for the embedding into multiple exponential spaces. Our main result is Theorem where we show that if one passes to unbounded domains, then, after the usual modification of the integrand in the Moser functional, the statement of the Concentration-Compactnes Principle...

On a Sobolev type inequality and its applications

Witold Bednorz (2006)

Studia Mathematica

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Assume ||·|| is a norm on ℝⁿ and ||·||⁎ its dual. Consider the closed ball T : = B | | · | | ( 0 , r ) , r > 0. Suppose φ is an Orlicz function and ψ its conjugate. We prove that for arbitrary A,B > 0 and for each Lipschitz function f on T, s u p s , t T | f ( s ) - f ( t ) | 6 A B ( 0 r ψ ( 1 / A ε n - 1 ) ε n - 1 d ε + 1 / ( n | B | | · | | ( 0 , 1 ) | ) T φ ( 1 / B | | f ( u ) | | ) d u ) , where |·| is the Lebesgue measure on ℝⁿ. This is a strengthening of the Sobolev inequality obtained by M. Talagrand. We use this inequality to state, for a given concave, strictly increasing function η: ℝ₊ → ℝ with η(0) = 0, a necessary and sufficient condition on...

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.

On uniqueness of distribution of a random variable whose independent copies span a subspace in L p

S. Astashkin, F. Sukochev, D. Zanin (2015)

Studia Mathematica

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Let 1 ≤ p < 2 and let L p = L p [ 0 , 1 ] be the classical L p -space of all (classes of) p-integrable functions on [0,1]. It is known that a sequence of independent copies of a mean zero random variable f L p spans in L p a subspace isomorphic to some Orlicz sequence space l M . We give precise connections between M and f and establish conditions under which the distribution of a random variable f L p whose independent copies span l M in L p is essentially unique.

On the inclusions of X Φ spaces

Seyyed Mohammad Tabatabaie, Alireza Bagheri Salec (2023)

Mathematica Bohemica

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We give some equivalent conditions (independent from the Young functions) for inclusions between some classes of X Φ spaces, where Φ is a Young function and X is a quasi-Banach function space on a σ -finite measure space ( Ω , 𝒜 , μ ) .

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 ) .

Multiparameter ergodic Cesàro-α averages

A. L. Bernardis, R. Crescimbeni, C. Ferrari Freire (2015)

Colloquium Mathematicae

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Net (X,ℱ,ν) be a σ-finite measure space. Associated with k Lamperti operators on L p ( ν ) , T , . . . , T k , n ̅ = ( n , . . . , n k ) k and α ̅ = ( α , . . . , α k ) with 0 < α j 1 , we define the ergodic Cesàro-α̅ averages n ̅ , α ̅ f = 1 / ( j = 1 k A n j α j ) i k = 0 n k i = 0 n j = 1 k A n j - i j α j - 1 T k i k T i f . For these averages we prove the almost everywhere convergence on X and the convergence in the L p ( ν ) norm, when n , . . . , n k independently, for all f L p ( d ν ) with p > 1/α⁎ where α = m i n 1 j k α j . In the limit case p = 1/α⁎, we prove that the averages n ̅ , α ̅ f converge almost everywhere on X for all f in the Orlicz-Lorentz space Λ ( 1 / α , φ m - 1 ) with φ ( t ) = t ( 1 + l o g t ) m . To obtain the result in the limit case we need...

On the Dirichlet Problem with Orlicz Boundary Data

Gabriella Zecca (2007)

Bollettino dell'Unione Matematica Italiana

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Let us consider a Young's function Φ : + + satisfying the Δ 2 condition together with its complementary function Ψ , and let us consider the Dirichlet problem for a second order elliptic operator in divergence form: { L u = 0 in  B u | B = f B the unit ball of n . In this paper we give a necessary and sufficient condition for the L ϕ -solvability of the problem, where L ϕ is the Orlicz Space generated by the function Φ . This means solvability for f L Φ in the sense of [5], [8], where the case Φ ( t ) = t p is treated. ...

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.