Potential theory (Mathematics)

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.

The Marchenko version of the s-wave many-channel
Bargmann potentials is first generalized to include bound
states and then employed to obtain the corresponding Theis
version. This is done with the expectation that, in contrast
to the Marchenko version, the Theis version can be
generalized to furnish many-channel Bargmann potentials for
arbitrarily high angular momenta.