Gradient regularity via rearrangements for $p$-Laplacian type elliptic boundary value problems

Andrea Cianchi; Vladimir G. Maz'ya

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An existence theorem for a strongly nonlinear elliptic problem in Musielak-Orlicz spaces

Abdelmoujib Benkirane, Fatimazahra Blali, Mohamed Sidi El Vally (2014)

Applicationes Mathematicae

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We prove an existence result for some class of strongly nonlinear elliptic problems in the Musielak-Orlicz spaces $W ¹ L_{φ} (Ω)$ , under the assumption that the conjugate function of φ satisfies the Δ₂-condition.

An inverse Sidon type inequality

S. Fridli (1993)

Studia Mathematica

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Sidon proved the inequality named after him in 1939. It is an upper estimate for the integral norm of a linear combination of trigonometric Dirichlet kernels expressed in terms of the coefficients. Since the estimate has many applications for instance in $L^{1}$ convergence problems and summation methods with respect to trigonometric series, newer and newer improvements of the original inequality has been proved by several authors. Most of them are invariant with respect to the rearrangement...

On a variant of the Hardy inequality between weighted Orlicz spaces

Agnieszka Kałamajska, Katarzyna Pietruska-Pałuba (2009)

Studia Mathematica

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Let M be an N-function satisfying the Δ₂-condition, and let ω, φ be two other functions, with ω ≥ 0. We study Hardy-type inequalities $\int_{ℝ ₊} M (ω (x) | u (x) |) e x p (- φ (x)) d x \leq C \int_{ℝ ₊} M (| u^{'} (x) |) e x p (- φ (x)) d x$ , where u belongs to some set of locally absolutely continuous functions containing $C ₀^{\infty} (ℝ ₊)$ . We give sufficient conditions on the triple (ω,φ,M) for such inequalities to be valid for all u from a given set . The set may be smaller than the set of Hardy transforms. Bounds for constants are also given, yielding classical Hardy inequalities with best constants. ...

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 $^{Ψ}$ .

The existence of solutions for elliptic systems with nonuniform growth

Y. Q. Fu (2002)

Studia Mathematica

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We study the Dirichlet problems for elliptic partial differential systems with nonuniform growth. By means of the Musielak-Orlicz space theory, we obtain the existence of weak solutions, which generalizes the result of Acerbi and Fusco [1].

Weighted inequalities for one-sided maximal functions in Orlicz spaces

Pedro Ortega Salvador (1998)

Studia Mathematica

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Let $M_{g}^{+}$ be the maximal operator defined by $M_{g}^{+} ⨍ (x) = s u p_{h > 0} (ʃ_{x}^{x + h} | ⨍ | g) / (ʃ_{x}^{x + h} g)$ , where g is a positive locally integrable function on ℝ. Let Φ be an N-function such that both Φ and its complementary N-function satisfy $Δ_{2}$ . We characterize the pairs of positive functions (u,ω) such that the weak type inequality $u (x \in ℝ | M_{g}^{+} ⨍ (x) > λ) \leq C / (Φ (λ)) ʃ_{ℝ} Φ (| ⨍ |) ω$ holds for every ⨍ in the Orlicz space $L_{Φ} (ω)$ . We also characterize the positive functions ω such that the integral inequality $ʃ_{ℝ} Φ (| M_{g}^{+} ⨍ |) ω \leq ʃ_{ℝ} Φ (| ⨍ |) ω$ holds for every $⨍ \in L_{Φ} (ω)$ . Our results include some already obtained for functions in $L^{p}$ and yield...

Strongly exposed points in Musielak-Orlicz sequence spaces endowed with Orlicz norm

Zhongrui Shi, Chunyan Liu (2011)

Banach Center Publications

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A criterion for strongly exposed points of the unit ball $B (l_{M})$ in Musielak-Orlicz sequence spaces $l_{M}$ equipped with Orlicz norm is given.

Reverse-Holder classes in the Orlicz spaces setting

E. Harboure, O. Salinas, B. Viviani (1998)

Studia Mathematica

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In connection with the $A_{ϕ}$ classes of weights (see [K-T] and [B-K]), we study, in the context of Orlicz spaces, the corresponding reverse-Hölder classes $R H_{ϕ}$ . We prove that when ϕ is $Δ_{2}$ and has lower index greater than one, the class $R H_{ϕ}$ coincides with some reverse-Hölder class $R H_{q}, q > 1$ . For more general ϕ we still get $R H_{ϕ} \subset A_{\infty} = ⋃_{q > 1} R H_{q}$ although the intersection of all these $R H_{ϕ}$ gives a proper subset of $⋂_{q > 1} R H_{q}$ .

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 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...

Orlicz spaces, α-decreasing functions, and the Δ₂ condition

Gary M. Lieberman (2004)

Colloquium Mathematicae

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We prove some quantitatively sharp estimates concerning the Δ₂ and ∇₂ conditions for functions which generalize known ones. The sharp forms arise in the connection between Orlicz space theory and the theory of elliptic partial differential equations.

Optimal integrability of the Jacobian of orientation preserving maps

Andrea Cianchi (1999)

Bollettino dell'Unione Matematica Italiana

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Dato un qualsiasi spazio invariante per riordinamenti $X (Ω)$ su un insieme aperto $Ω \subset R^{n}$ , si determina il più piccolo spazio invariante per riordinamenti $Y (Ω)$ con la proprietà che se $u : Ω \to R^{n}$ è una applicazione che mantiene l'orientamento e ${|D u|}^{n} \in X (Ω)$ , allora $det D u$ appartiene localmente a $Y (Ω)$ .

On the k-convexity of the Besicovitch-Orlicz space of almost periodic functions with the Orlicz norm

Fazia Bedouhene, Mohamed Morsli (2007)

Colloquium Mathematicae

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Boulahia and the present authors introduced the Orlicz norm in the class $B^{ϕ}$ -a.p. of Besicovitch-Orlicz almost periodic functions and gave several formulas for it; they also characterized the reflexivity of this space [Comment. Math. Univ. Carolin. 43 (2002)]. In the present paper, we consider the problem of k-convexity of $B^{ϕ}$ -a.p. with respect to the Orlicz norm; we give necessary and sufficient conditions in terms of strict convexity and reflexivity.

Geometry of Orlicz spaces

Chen Shutao

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

Existence Theorems for the Dirichlet Elliptic Inclusion Involving Exponential-Growth-Type Multivalued Right-Hand Side

Hôǹg Thái Nguyêñ, Dariusz Pączka (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

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We present two existence results for the Dirichlet elliptic inclusion with an upper semicontinuous multivalued right-hand side in exponential-type Orlicz spaces involving a vector Laplacian, subject to Dirichlet boundary conditions on a domain Ω⊂ ℝ². The first result is obtained via the multivalued version of the Leray-Schauder principle together with the Nakano-Dieudonné sequential weak compactness criterion. The second result is obtained by using the nonsmooth variational technique...

On linear functionals in Hardy-Orlicz spaces, I

R. Leśniewicz (1973)

Studia Mathematica

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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.

Copies of $l^{1}$ and $c_{o}$ in Musielak-Orlicz sequence spaces

Ghassan Alherk, Henryk Hudzik (1994)

Commentationes Mathematicae Universitatis Carolinae

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Criteria in order that a Musielak-Orlicz sequence space $l^{Φ}$ contains an isomorphic as well as an isomorphically isometric copy of $l^{1}$ are given. Moreover, it is proved that if $Φ = (Φ_{i})$ , where $Φ_{i}$ are defined on a Banach space, $X$ does not satisfy the $δ_{2}^{o}$ -condition, then the Musielak-Orlicz sequence space $l^{Φ} (X)$ of $X$ -valued sequences contains an almost isometric copy of $c_{o}$ . In the case of $X = I R$ it is proved also that if $l^{Φ}$ contains an isomorphic copy of $c_{o}$ , then $Φ$ does not satisfy the $δ_{2}^{o}$ -condition. These results...

Displaying similar documents to “Gradient regularity via rearrangements for p -Laplacian type elliptic boundary value problems”

Displaying similar documents to “Gradient regularity via rearrangements for $p$ -Laplacian type elliptic boundary value problems”