Supervisors info:
Μπαρμπάτης Γεράσιμος Αναπληρωτής Καθηγητής (επιβλέπων), Κατάβολος Αριστείδης Καθηγητής, Στρατής Ιωάννης Καθηγητής
Summary:
In this thesis, we study the boundedness of Singular Integral Operators (SIOs)
and
in particular the Riesz transform on Lp(Rd), but also on Riemannian manifolds.
In particular, we first prove that every bounded SIO on L^q(R^d), for some 1 <
q , that satisfies a certain Hormander property, is weakly L^1 bounded.Thus,
the Marcinkiewicz interpolation theorem implies that these SIOs are bounded on
Lp(Rd),for 1 < p< q. Since the Riesz transforms are skewadjoint and satisfy the
above properties, they are L^pbounded, for every 1 < p . Secondly, we prove
that the L^p boundedness of the Riesz transforms can be extended to a
reasonable class of complete noncompact
Riemannian manifolds. Particularly, we give positive results for 1 < p <2
under very weak assumptions, namely the doubling volume property and an
ondiagonal heat kernel estimate. We also prove that the result cannot hold for
p > 2 under the same assumptions.
Keywords:
Marcinkiewicz, Calderon Zygmund, Riesz , Heat kernel, Riemann
