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Sobolev Inequalities
- 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 |
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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 |
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Subject(s): |
Sobolev spaces Inequalities (Mathematics) Nonlinear theories Potential theory (Mathematics) |
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Held by: | Florida Atlantic University Libraries | |
Persistent Link to This Record: | http://purl.flvc.org/fau/fd/FA00000863 | |
Sublocation: | Digital Library | |
Use and Reproduction: | Copyright © is held by the author with permission granted to Florida Atlantic University to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder. | |
Use and Reproduction: | http://rightsstatements.org/vocab/InC/1.0/ | |
Host Institution: | FAU | |
Is Part of Series: | Florida Atlantic University Digital Library Collections. |