Some functional inequalities and spectral properties of metric measure spaces with curvature bounded below

[from the introduction]: The aim of this thesis is to study metric measure spaces with a synthetic notion of Ricci curvature bounded below. We study them from the point of view of Sobolev/Nash type functional inequalities in the non-compact case, and from the point of view of spectral analysis in the compact case. The heat kernel links the two cases: in the first one, the goal is to get new estimates on the heat kernel of some associated weighted structure; in the second one, the heat kernel is the basic tool to establish our results. The topic of synthetic Ricci curvature bounds has known a constant development over the past few years. In this introduction, we shall give some historical account on this theory, before explaining in few words the content of this work. The letter K will refer to an arbitrary real number and N will refer to any finite number greater or equal than 1.