Displaying similar documents to “Optimal embeddings of generalized homogeneous Sobolev spaces”

Optimal Sobolev imbedding spaces

Ron Kerman, Luboš Pick (2009)

Studia Mathematica

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This paper continues our study of Sobolev-type imbedding inequalities involving rearrangement-invariant Banach function norms. In it we characterize when the norms considered are optimal. Explicit expressions are given for the optimal partners corresponding to a given domain or range norm.

Optimal domains for the kernel operator associated with Sobolev's inequality

Guillermo P. Curbera, Werner J. Ricker (2003)

Studia Mathematica

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Refinements of the classical Sobolev inequality lead to optimal domain problems in a natural way. This is made precise in recent work of Edmunds, Kerman and Pick; the fundamental technique is to prove that the (generalized) Sobolev inequality is equivalent to the boundedness of an associated kernel operator on [0,1]. We make a detailed study of both the optimal domain, providing various characterizations of it, and of properties of the kernel operator when it is extended to act in its...

Dimension-invariant Sobolev imbeddings

Miroslav Krbec, Hans-Jürgen Schmeisser (2011)

Banach Center Publications

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We survey recent dimension-invariant imbedding theorems for Sobolev spaces.

A sharp iteration principle for higher-order Sobolev embeddings

Andrea Cianchi, Luboš Pick, Lenka Slavíková (2014)

Banach Center Publications

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We survey results from the paper [CPS] in which we developed a new sharp iteration method and applied it to show that the optimal Sobolev embeddings of any order can be derived from isoperimetric inequalities. We prove thereby that the well-known link between first-order Sobolev embeddings and isoperimetric inequalities translates to embeddings of any order, a fact that had not been known before. We show a general reduction principle that reduces Sobolev type inequalities of any order...

Variable Sobolev capacity and the assumptions on the exponent

Petteri Harjulehto, Peter Hästö, Mika Koskenoja, Susanna Varonen (2005)

Banach Center Publications

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In a recent article the authors showed that it is possible to define a Sobolev capacity in variable exponent Sobolev space. However, this set function was shown to be a Choquet capacity only under certain assumptions on the variable exponent. In this article we relax these assumptions.

Some new spaces of Besov and Triebel-Lizorkin type on homogeneous spaces

Yongsheng Han, Dachun Yang (2003)

Studia Mathematica

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New norms for some distributions on spaces of homogeneous type which include some fractals are introduced. Using inhomogeneous discrete Calderón reproducing formulae and the Plancherel-Pólya inequalities on spaces of homogeneous type, the authors prove that these norms give a new characterization for the Besov and Triebel-Lizorkin spaces with p, q > 1 and can be used to introduce new inhomogeneous Besov and Triebel-Lizorkin spaces with p, q ≤ 1 on spaces of homogeneous type. Moreover,...

Optimal domains for kernel operators on [0,∞) × [0,∞)

Olvido Delgado (2006)

Studia Mathematica

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Let T be a kernel operator with values in a rearrangement invariant Banach function space X on [0,∞) and defined over simple functions on [0,∞) of bounded support. We identify the optimal domain for T (still with values in X) in terms of interpolation spaces, under appropriate conditions on the kernel and the space X. The techniques used are based on the relation between linear operators and vector measures.

From the Prékopa-Leindler inequality to modified logarithmic Sobolev inequality

Ivan Gentil (2008)

Annales de la faculté des sciences de Toulouse Mathématiques

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We develop in this paper an improvement of the method given by S. Bobkov and M. Ledoux in [BL00]. Using the Prékopa-Leindler inequality, we prove a modified logarithmic Sobolev inequality adapted for all measures on ℝ n , with a strictly convex and super-linear potential. This inequality implies modified logarithmic Sobolev inequality, developed in [GGM05, GGM07], for all uniformly strictly convex potential as well as the Euclidean logarithmic Sobolev inequality.

Mixed norms and Sobolev type inequalities

V. I. Kolyada (2006)

Banach Center Publications

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We study mixed norm spaces that arise in connection with embeddings of Sobolev and Besov spaces. We prove Sobolev type inequalities in terms of these mixed norms. Applying these results, we obtain optimal constants in embedding theorems for anisotropic Besov spaces. This gives an extension of the estimate proved by Bourgain, Brezis and Mironescu for isotropic Besov spaces.

An embedding theorem for Sobolev type functions with gradients in a Lorentz space

Alireza Ranjbar-Motlagh (2009)

Studia Mathematica

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The purpose of this paper is to prove an embedding theorem for Sobolev type functions whose gradients are in a Lorentz space, in the framework of abstract metric-measure spaces. We then apply this theorem to prove absolute continuity and differentiability of such functions.

Generalized homogeneous Besov spaces and their applications

Mejjaoli, Hatem (2012)

Serdica Mathematical Journal

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2010 Mathematics Subject Classification: Primary 35L05. Secondary 46E35, 35J25, 22E30. In this paper we define the homogeneous Besov spaces associated with the Dunkl operators on R^d, and we give a complete analysis on these spaces and same applications.