Sobolev Inequalities

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.