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Sharp Summability of Functions From Orlicz-Sobolev Spaces

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Abstract

Recent years have witnessed an increasing interest of researchers working on function spaces, and on related fields, about optimal embeddings of Sobolev type, as demonstrated by a number of papers on this topic. One of the ancestors of these kind of results may be considered a sharpened version of the classical Sobolev inequality, independently proved by O’Neil [23] and by Peetre [25], which can be stated as follows. LetG be an open subset of ℝn,n> 2, and let Wo l’P(G), 1 <p < ∞, be the first order Sobolev space of those real-valued weakly differentiable functions inGdecaying to 0 on∂G whose gradient belongs toL p(G). If 1 <p < n, then a constantC exists such that

$$ \left\| u \right\|_{L^{p^* ,p} (G)} \leqslant C\left\| {\left| \nabla \right.} \right.\left. {\left. u \right|} \right\|_{L^p (G)}$$
((1))

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Authors and Affiliations

  1. Dipartimento di Matematica e Applicazioni per l’Architettura, Università di Firenze, Piazza Ghiberti 27, Firenze, 50122, Italy

    Andrea Cianchi

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  1. Andrea Cianchi
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Editors and Affiliations

  1. Institute of Mathematics, Friedrich-Schiller-University, Jena, 07740, Germany

    Dorothee Haroske , Thomas Runst  & Hans-Jürgen Schmeisser ,  & 

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Cianchi, A. (2003). Sharp Summability of Functions From Orlicz-Sobolev Spaces. In: Haroske, D., Runst, T., Schmeisser, HJ. (eds) Function Spaces, Differential Operators and Nonlinear Analysis. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8035-0_14

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  • DOI: https://doi.org/10.1007/978-3-0348-8035-0_14

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9414-2

  • Online ISBN: 978-3-0348-8035-0

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