Displaying similar documents to “Trudinger's inequality for double phase functionals with variable exponents”

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.

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.

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...

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...

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 .

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.

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 ) .

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.

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...

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...

Weighted local Orlicz-Hardy spaces with applications to pseudo-differential operators

Dachun Yang, Sibei Yang

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Let Φ be a concave function on (0,∞) of strictly critical lower type index p Φ ( 0 , 1 ] and ω A l o c ( ) (the class of local weights introduced by V. S. Rychkov). We introduce the weighted local Orlicz-Hardy space h ω Φ ( ) via the local grand maximal function. Let ρ ( t ) t - 1 / Φ - 1 ( t - 1 ) for all t ∈ (0,∞). We also introduce the BMO-type space b m o ρ , ω ( ) and establish the duality between h ω Φ ( ) and b m o ρ , ω ( ) . Characterizations of h ω Φ ( ) , including the atomic characterization, the local vertical and the local nontangential maximal function characterizations, are...

The σ -property in C ( X )

Anthony W. Hager (2016)

Commentationes Mathematicae Universitatis Carolinae

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The σ -property of a Riesz space (real vector lattice) B is: For each sequence { b n } of positive elements of B , there is a sequence { λ n } of positive reals, and b B , with λ n b n b for each n . This condition is involved in studies in Riesz spaces of abstract Egoroff-type theorems, and of the countable lifting property. Here, we examine when “ σ ” obtains for a Riesz space of continuous real-valued functions C ( X ) . A basic result is: For discrete X , C ( X ) has σ iff the cardinal | X | < 𝔟 , Rothberger’s bounding number. Consequences...

Generalized versions of Ilmanen lemma: Insertion of C 1 , ω or C loc 1 , ω functions

Václav Kryštof (2018)

Commentationes Mathematicae Universitatis Carolinae

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We prove that for a normed linear space X , if f 1 : X is continuous and semiconvex with modulus ω , f 2 : X is continuous and semiconcave with modulus ω and f 1 f 2 , then there exists f C 1 , ω ( X ) such that f 1 f f 2 . Using this result we prove a generalization of Ilmanen lemma (which deals with the case ω ( t ) = t ) to the case of an arbitrary nontrivial modulus ω . This generalization (where a C l o c 1 , ω function is inserted) gives a positive answer to a problem formulated by A. Fathi and M. Zavidovique in 2010.

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...

𝒞 k -regularity for the ¯ -equation with a support condition

Shaban Khidr, Osama Abdelkader (2017)

Czechoslovak Mathematical Journal

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Let D be a 𝒞 d q -convex intersection, d 2 , 0 q n - 1 , in a complex manifold X of complex dimension n , n 2 , and let E be a holomorphic vector bundle of rank N over X . In this paper, 𝒞 k -estimates, k = 2 , 3 , , , for solutions to the ¯ -equation with small loss of smoothness are obtained for E -valued ( 0 , s ) -forms on D when n - q s n . In addition, we solve the ¯ -equation with a support condition in 𝒞 k -spaces. More precisely, we prove that for a ¯ -closed form f in 𝒞 0 , q k ( X D , E ) , 1 q n - 2 , n 3 , with compact support and for ε with 0 < ε < 1 there...

L p , q spaces

Joseph Kupka

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CONTENTS1. Introduction...................................................................................................... 52. Notation and basic terminology........................................................................... 73. Definition and basic properties of the L p , q spaces................................. 114. Integral representation of bounded linear functionals on L p , q ( B ) ........ 235. Examples in L p , q theory...................................................................................

Sum-product theorems and incidence geometry

Mei-Chu Chang, Jozsef Solymosi (2007)

Journal of the European Mathematical Society

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In this paper we prove the following theorems in incidence geometry. 1. There is δ > 0 such that for any P 1 , , P 4 , and Q 1 , , Q n 2 , if there are n ( 1 + δ ) / 2 many distinct lines between P i and Q j for all i , j , then P 1 , , P 4 are collinear. If the number of the distinct lines is < c n 1 / 2 then the cross ratio of the four points is algebraic. 2. Given c > 0 , there is δ > 0 such that for any P 1 , P 2 , P 3 2 noncollinear, and Q 1 , , Q n 2 , if there are c n 1 / 2 many distinct lines between P i and Q j for all i , j , then for any P 2 { P 1 , P 2 , P 3 } , we have δ n distinct lines between P and Q j . 3. Given...