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Sobolev Inequalities

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Date Issued:
2007
Summary:
Recently a rich theory of Sobolev spaces on metric spaces has been developed. It. turas out that many relevant results from the classical theory have their counterparts in the mcnic setting ( cf. [P. Hajlasz and P. Koskela. Sobokv met Poincare), Mern. Arner. Math. Soc. 145 (2000), no. 6888, x+101pp]). In this thesis we prove sharp Sobolev inequalities in the context of metric spaces. Our approach is ba....,ed on two recent papers, [J. Baster·o and M. Milman and F. Ruiz, A note on L(oc, q) spaces and Sobolev embeddings, Indiana Univ. Math. J. 52 (2003), no. 5, 1215- 1230] and [J. Martfn and M. Milman and E. Pustylnik, Sobolev inequalities: symmetrization and self improvement via truncation, to appear in J. Funct. Anal.]. These authors establish sharp Sobolev embeddings in the Euclidean setting using symmetrization. Using suitable maximal operators and covering lemmas we show that these symmetrization inequalities of Bastero-Milman-Ruiz remain valid m the metric setting. We also show that the symmetrization by truncation method of Martfn-Milman-Pustylnik can be implemented in our generalized setting. Furthermore we also show that our methods can be adapted to deal with non-doubling measures.
Title: Sobolev Inequalities.
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Name(s): Kalis, Jan
Florida Atlantic University, Degree grantor
Milman, Mario, Thesis advisor
Charles E. Schmidt College of Science
Department of Mathematical Sciences
Type of Resource: text
Genre: Electronic Thesis Or Dissertation
Date Created: 2007
Date Issued: 2007
Publisher: Florida Atlantic University
Place of Publication: Boca Raton, Fla.
Physical Form: application/pdf
Extent: 68 p.
Language(s): English
Summary: Recently a rich theory of Sobolev spaces on metric spaces has been developed. It. turas out that many relevant results from the classical theory have their counterparts in the mcnic setting ( cf. [P. Hajlasz and P. Koskela. Sobokv met Poincare), Mern. Arner. Math. Soc. 145 (2000), no. 6888, x+101pp]). In this thesis we prove sharp Sobolev inequalities in the context of metric spaces. Our approach is ba....,ed on two recent papers, [J. Baster·o and M. Milman and F. Ruiz, A note on L(oc, q) spaces and Sobolev embeddings, Indiana Univ. Math. J. 52 (2003), no. 5, 1215- 1230] and [J. Martfn and M. Milman and E. Pustylnik, Sobolev inequalities: symmetrization and self improvement via truncation, to appear in J. Funct. Anal.]. These authors establish sharp Sobolev embeddings in the Euclidean setting using symmetrization. Using suitable maximal operators and covering lemmas we show that these symmetrization inequalities of Bastero-Milman-Ruiz remain valid m the metric setting. We also show that the symmetrization by truncation method of Martfn-Milman-Pustylnik can be implemented in our generalized setting. Furthermore we also show that our methods can be adapted to deal with non-doubling measures.
Identifier: FA00000863 (IID)
Degree granted: Dissertation (Ph.D.)--Florida Atlantic University, 2007.
Collection: FAU Electronic Theses and Dissertations Collection
Note(s): Includes bibliography.
Charles E. Schmidt College of Science
Subject(s): Sobolev spaces
Inequalities (Mathematics)
Nonlinear theories
Potential theory (Mathematics)
Held by: Florida Atlantic University Libraries
Persistent Link to This Record: http://purl.flvc.org/fau/fd/FA00000863
Sublocation: Digital Library
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Host Institution: FAU
Is Part of Series: Florida Atlantic University Digital Library Collections.