main-content

This book provides an introduction to some aspects of the flourishing field of nonsmooth geometric analysis. In particular, a quite detailed account of the first-order structure of general metric measure spaces is presented, and the reader is introduced to the second-order calculus on spaces – known as RCD spaces – satisfying a synthetic lower Ricci curvature bound. Examples of the main topics covered include notions of Sobolev space on abstract metric measure spaces; normed modules, which constitute a convenient technical tool for the introduction of a robust differential structure in the nonsmooth setting; first-order differential operators and the corresponding functional spaces; the theory of heat flow and its regularizing properties, within the general framework of “infinitesimally Hilbertian” metric measure spaces; the RCD condition and its effects on the behavior of heat flow; and second-order calculus on RCD spaces. The book is mainly intended for young researchers seeking a comprehensive and fairly self-contained introduction to this active research field. The only prerequisites are a basic knowledge of functional analysis, measure theory, and Riemannian geometry.

### Chapter 1. Preliminaries

Abstract
In this chapter we introduce several classic notions that will be needed in the sequel. Namely, in Sect. 1.1 we review the basics of measure theory, with a particular accent on the space $$L^0({\mathbf {\mathfrak {m}}})$$ of Borel functions considered up to $${\mathbf {\mathfrak {m}}}$$-almost everywhere equality (see Sect. 1.1.2); in Sect. 1.2 we discuss about continuous, absolutely continuous and geodesic curves on metric spaces; in Sect. 1.3 we collect the most important results about Bochner integration. Some functional analytic tools will be treated in Appendix A.
Nicola Gigli, Enrico Pasqualetto

### Chapter 2. Sobolev Calculus on Metric Measure Spaces

Abstract
Several different approaches to the theory of weakly differentiable functions over abstract metric measure spaces made their appearance in the literature throughout the last twenty years. Amongst them, we shall mainly follow the one (based upon the concept of test plan) that has been proposed by Ambrosio, Gigli and Savaré. The whole Sect. 2.1 is devoted to the definition of such notion of Sobolev space W 1, 2(X) and to its most important properties.
Nicola Gigli, Enrico Pasqualetto

### Chapter 3. The Theory of Normed Modules

Abstract
This chapter is devoted to the study of the so-called normed modules over metric measure spaces. These represent a tool that has been introduced by Gigli in order to build up a differential structure on nonsmooth spaces. In a few words, an $$L^2({{\mathfrak {m}}})$$-normed $$L^\infty ({{\mathfrak {m}}})$$-module is a generalisation of the concept of ‘space of 2-integrable sections of some measurable bundle’; it is an algebraic module over the commutative ring $$L^\infty ({{\mathfrak {m}}})$$ that is additionally endowed with a pointwise norm operator. This notion, its basic properties and some of its technical variants constitute the topics of Sect. 3.1.
Nicola Gigli, Enrico Pasqualetto

### Chapter 4. First-Order Calculus on Metric Measure Spaces

Abstract
In this chapter we develop a first-order differential structure on general metric measure spaces. First of all, the key notion of cotangent module is obtained by combining the Sobolev calculus (discussed in Chap. 2) with the theory of normed modules (described in Chap. 3). The elements of the cotangent module L 2(T X), which are defined and studied in Sect. 4.1, provide a convenient abstraction of the concept of ‘1-form on a Riemannian manifold’.
Nicola Gigli, Enrico Pasqualetto

### Chapter 5. Heat Flow on Metric Measure Spaces

Abstract
In order to develop a second-order differential calculus on spaces with curvature bounds we need to make use of the regularising effects of the heat flow, to which this chapter is dedicated.
Nicola Gigli, Enrico Pasqualetto

### Chapter 6. Second-Order Calculus on RCD Spaces

Abstract
In this conclusive chapter we introduce the class of those metric measure spaces that satisfy the Riemannian curvature-dimension condition, briefly called RCD spaces, and we develop a thorough second-order differential calculus over these structures.
Nicola Gigli, Enrico Pasqualetto