Author Type

Graduate Student

Date of Award

2007

Document Type

Dissertation

Publication Status

Version of Record

Submission Date

October 2026

Thesis/Dissertation Advisor [Chair]

Milman, Mario

Abstract

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.

Extent

68 p.

Comments

Includes bibliography.|Dissertation (Ph.D.)--Florida Atlantic University, 2007.|FAU Electronic Theses and Dissertations Collection|Charles E. Schmidt College of Science|Includes bibliography.

Identifier

FA00000863

Format

pdf

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