Inequalities (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.

Assume that {φn} is an orthonormal uniformly bounded (ONB) sequence of complex-valued functions de ned on a measure space (Ω,Σ,µ), and f ∈ L1(Ω,Σ,µ). Let
be the Fourier coefficients of f with respect to {φn} .
R.E.A.C. Paley proved a theorem connecting the Lp-norm of f with a related norm of the sequence {cn}. Hardy and Littlewood subsequently proved that Paley’s result is best possible within its context. Their results were generalized by Dikarev, Macaev, Askey, Wainger, Sagher, and later by Tikhonov, Li yand, Booton and others.The present work continues the generalization of these results.