Some Local Multipliers Associated with the Ornstein-Uhlenbeck Operator

Abstract

This thesis combines ideas from [19] and [21] to introduce two main spectral multipliers results for the Ornstein Uhlenbeck operator in Section 4. Chapter 1 introduces basic de nitions and properties of Markov semigroups and shows how they relate to the Ornstein Uhlenbeck operator. Lots of researchers study the hy- percontractivity result via Markov segmigroups. Chapter 2 describes the notions of weighted Nash inequalities and show how to deduce a hypercontrativity result (Theorem 2.7) from these inequalities based on paper [13]. A brief introduction to the Spectral theory and the main theorem of paper [19] is given in Section 3.