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Sobolev Inequalities
Title
Sobolev Inequalities.
Creator
Kalis, Jan, Florida Atlantic University, Milman, Mario, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description
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)...
Show moreRecently 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.
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Date Issued
2007
PURL
http://purl.flvc.org/fau/fd/FA00000863
Subject Headings
Sobolev spaces, Inequalities (Mathematics), Nonlinear theories, Potential theory (Mathematics)
Format
Document (PDF)