Diffusion Generators and Associated Inequalities for Powers of the Laplacian Operator

Abstract

In this paper, we study the properties of semigroup operators and the curvature dimension inequalities associated with them, especially under the consideration of the Diffusion property. We are also interested in the existence of certain functional inequalities, especially spectral gap, Sobolev, and logarithmic Sobolev inequalities. We will begin by discussing the basic properties of semigroups and their infinitesimal generators and also a few important results providing some intuition for and equivalent statements of the definition of the CD-inequality (in the Bakry-Emery sense). We will then provide arguments for a couple of important results relating these CD-inequalities to spectral gap, Sobolev, and log-Sobolev inequalities. After establishing this theory, we will restrict ourselves to the analysis of a few well-known generators: the Laplacian (generating the heat semigroup and describing Brownian motion) on $\mathbb{R}^n$ and on some manifold $(M,g)$ and the Ornstein-Uhlenbeck generator (on an invariant Gaussian measure). We will then attempt to describe the semigroup generated by the biharmonic operator, showing that CD-inequalities are not upheld in any dimension $n$ for all measurable and bounded functions $f \in \mathbb{R}^n$. Even so we show that it is possible, at least for some specific measures with weights satisfying an $A_p$ condition, to arrive at a Sobolev inequality for the biharmonic operator. We end with a discussion of the fractional Laplacian operator and pose a new set of potential operators and curvature dimension inequalities which might be used to find functional inequalities associated with the fractional case.