Unit:
Κατεύθυνση Θεωρητικά ΜαθηματικάLibrary of the School of Science
Supervisors info:
Απόστολος Γιαννόπουλος, Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ (επιβλέπων)
Αριστείδης Κατάβολος, Ομότιμος Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ
Αντώνιος Τσολομύτης, Καθηγητής, Τμήμα Μαθηματικών, Πανεπιστήμιο Αιγαίου
Original Title:
Ενισχυμένες ανισότητες Hölder και αντίστροφες Hölder για Gaussian τυχαία διανύσματα
Translated title:
Improved Holder and inverse Holder inequalities for Gaussian random vectors
Summary:
The main goal of this thesis is to present results of Chen, Dafnis and Paouris which
provide algebraic criteria that ensure improved Hölder and inverse Hölder inequalities
for products of functions of Gaussian random vectors with arbitrary covariance structure.
We will also see that these inequalities generalize classical results such as the Prékopa-
Leindler inequality, the Brascamp-Lieb inequality and its inverse Barthe inequality, and
we will study the geometry of eligible exponents for these inequalities. Finally, we will
see how one can use these inequalities to prove the Ehrhard-Borell inequality.
More specifically at thesis:
• In the 3rd chapter we present two proofs by Dafni, Paouris and Chen for enhanced Hölder and inverse Hölder inequalities, the first is done using Gaussian integration by parts, while the second is using the Ornstein-Uhlenbeck semigroup.
• In the 4th chapter we study interconnections of the basic Theorem with known inequalities. Next we will study some geometrical properties of selectable exponents of the Basic Theorem, we will also show that it generalizes Gaussian hypercontractibility and its inverse form.
•In the 5th chapter we will present Barthe's Lemma as well as the proof he gave, we also present how this Lemma can be generalized from the Theorem of Dafni, Paouris and Chen.
• In chapter 6 we prove an exact inequality for the Gaussian moments of logarithmically concave or logarithmically convex functions and then prove a stability result for the logarithmic Sobolev inequality.
• In chapter 7 we present a proof of Ehrhad's inequality due to Paouris and Neeman, who construct a quantity that is monotonic along the Ornstein-Uhlenbeck semigroup.
Main subject category:
Science
Keywords:
Holder Inequalities, Inverse Holder inequalities, Logarithmic Sobolev inequality, Ehrhard inequality
