Supervisors info:
Γιαννόπουλος Απόστολος, Καθηγητής του Τμήματος Μαθηματικών, ΕΚΠΑ (επιβλέπων)
Γατζούρας Δημήτριος, Καθηγητής του Τμήματος Μαθηματικών, ΕΚΠΑ
Μπαρμπάτης Γεράσιμος, Καθηγητής του Τμήματος Μαθηματικών, ΕΚΠΑ
Summary:
In this master thesis, we are studying functional inequalities with tools coming from Markov semigroup theory.Firstly, there is an extensive introduction in the Markov semigroup theory, while we define Markov triples which is the frame of our work in the next chapters.
In the beginning of the main body of this thesis, we present Sobolev, logarithmic entropy-energy and Nash inequalities and the interactions among them.In the next chapter, we begin with some ultracontractivity results, where we prove the equivalence between
Sobolev inequalities and uniform heat kernel bounds.Furthermore, we are studying the
stability of Sobolev inequalities in the product spaces of Markov triples and Sobolev
inequalities under Lipschitz functions.
Subsequently, we introduce local Sobolev inequalities.We prove local dimensional logarithmic Sobolev inequality and the equivalence of curvature-dimension condition with hypercontractivity properties.
In the next chapter, we prove Sobolev, log-Sobolev and Poincare inequalities under the curvature-dimension condition.Moreover, we are studying the conformal invariance of Sobolev inequalities and as a result we are led to the optimal Sobolev inequalities in sphere, Euclidean and Hyperbolic space.
In the last chapter, we are studying some recent techniques in order to prove Gagliardo-Nirenberg inequalities.Furthermore, we emphasize in a new family of interpolation inequalities called Sobolev-Kantorovich, based on the work of Ledoux.
Keywords:
Geometric analysis, Sobolev inequalities, Markov semigroup, functional inequalities
