Sobolev and Bounded Variation Functions on Metric Measure Spaces Luigi Ambrosio, Roberta Ghezzi Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy [email protected]. Institut de Math´ematiques de Bourgogne, UBFC, 9 Avenue Alain Savary, 21078 Dijon, France [email protected]. Contents Chapter 1. Introduction 1 1. History 1 2. Motivations 3 3. Examples of metric measure spaces 4 Chapter 2. H-Sobolev space and first tools of differential calculus 9 1. Relaxed slope and Cheeger energy 9 2. Elements of differential calculus 11 3. Reminders of convex analysis 14 4. Laplacian and integration by parts formula 15 5. Heat flow in (X; d; m) 16 Chapter 3. The Lagrangian (Beppo Levi) approach 19 1. Basic tools 19 2. The metric case 20 3. p-test plans and their relation with p-Modulus 22 4. The inclusion H1;p(X; d; m) ⊂ BL1;p(X; d; m) 24 5. Equivalence between H-space and BL-space 25 Chapter 4. Sobolev spaces via integration by parts 37 1. Vector fields 37 2. W 1;p-space and the inclusion H1;p(X; d; m) ⊂ W 1;p(X; d; m) 38 3. The inclusion W 1;p(X; d; m) ⊂ BL1;p(X; d; m) 40 Chapter 5. Functions of bounded variation 43 1. The spaces BV∗(X; d; m) and BVBL(X; d; m) 43 2. Structure of the perimeter measure 45 Bibliography 53 CHAPTER 1 Introduction These notes reflect, with minor modification and updates, the lectures given by the first author in occasion of the Trimester in Geometry, Analysis and Dynamics on Sub- Riemannian Manifolds held at Institut Henri Poincar´ein Paris, September 2014. Later on this series of lectures has been in part repeated by the first author within the Research Term in Analysis and Geometry in Metric Spaces at Instituto de Ciencias Matem´aticasin Madrid, May 2015. The style of the presentation is similar to the one of the lectures, in that we aim at the illustration of the key ideas and of the fundamental technical concepts, not looking for the most general statements, whose proofs are occasionally cumbersome. The references on the quite broad topic of Analysis in Metric Spaces do not pretend to be exhaustive, even looking only at the Sobolev theory; nevertheless we think that these lectures provide a good entry point to the literature and to the state of the art on this topic. 1. History n n n Consider the classical Euclidean space (R ; |·|; L ), where L denotes the n-dimensional Lebesgue measure. In this context one can give three definitions of Sobolev spaces, that we now know to be equivalent. 1 n 1 n Let 1 < p < 1 and denote by Cc (R ) the vector space of C functions on R with compact support. Definition 1.1 (W -definition via integration by parts). 1;p n n p n p n n W (R ) := f 2 L (R ) j 9 g 2 L (R ; R ) such that 1 n @' o 8 ' 2 Cc (R ); f dx = − 'gidx; i = 1; : : : ; n : ˆRn @xi ˆRn The vector g in the definition above, called weak gradient, is unique and denoted by rf. The W 1;p norm is then defined by p p1=p kfkW 1;p := kfkp + krfkp : Definition 1.2 (Sobolev definition or H-definition). W 1;p( n) 1;p n 1 n R H (R ) := Cc (R ) ; 1;p n 1 n 1;p n i.e., the Sobolev space H (R ) is the closure of Cc (R ) with respect to the W (R )-norm. Let us recall how the equivalence between Definitions 1.1 and 1.2 can be proved. 1;p n 1;p n Theorem 1.1. H (R ) = W (R ). 1 2 Introduction 1;p n 1;p n Sketch of the proof. The inclusion H (R ) ⊂ W (R ) is rather obvious, once 1;p n 1;p n the notion of convergence in W (R ) is well understood. Indeed, let f 2 H (R ) and let 1 n (fk) ⊂ Cc (R ) be an approximating sequence of f, i.e., lim kfk − fkW 1;p( n) = 0: (1) k!1 R 1 n Fix any ' 2 Cc (R ). Since fk is smooth and ' has compact support, using the integration by parts formula we obtain @' @fk fk dx = − ' dx: (2) ˆRn @xi ˆRn @xi p n Thanks to (1), the sequence (fk) converges to f strongly in L (R ) and there exists g 2 Lp( n; n) such that @fk converges to g strongly in Lp( n). Thus formula (2) goes to R R @xi i R the limit and we get that gi is indeed the weak derivative of f with respect to xi in the sense of distributions. (In other words, integration by parts formula is preserved when approximating by smooth functions). The opposite inclusion is due to Meyers and Serrin [48]: even in general domains Ω, their result is that C1(Ω) \ W 1;p(Ω) is dense in W 1;p(Ω)) and it is based on an approximation 1;p n 1 technique using mollifiers. Pick f 2 W (R ) and a sequence fρg>0 of mollifiers . Then 1 n 1;p n f ∗ ρ 2 C (R ) \ W (R ). Multiplying f ∗ ρ by a cut-off function and using a diagonal argument, one can build up a sequence of smooth functions with compact support converging 1;p n to f strongly in W (R ). Remark 1.1. In connection with sub-Riemannian geometry, a Meyers{Serrin's type result was proved by Franchi, Serapioni and Serra Cassano in [29] for metric measure n spaces associated with systems of vector fields in R satisfying mild hypotheses. For the definition of Sobolev space in this context, see Example 3.4 and references therein. n Definition 1.3 (Beppo Levi definition). Given a point x 2 R and an index i = 1; : : : ; n n−1 we denote byx ^i the point (x1; : : : ; xi−1; xi+1; : : : ; xn) 2 R . Then, we define 1;p n n n n−1 n−1 BL (R ) := f : R ! R j 8 i = 1; : : : ; n and at L -a.e.x ^i 2 R the map t 7! f(x1; : : : ; xi−1; t; xi+1; : : : ; xn) is absolutely continuous and p p @f o jfj + dt dx^i < 1 : ˆRn−1 ˆR @xi In the case n = p = 2, Definition 1.3 appeared much earlier than Definitions 1.1 and 1.2 in the work [45] by Beppo Levi, where the author constructs the space above in order to solve a specific Dirichlet problem in two-dimensional domains. Definition 1.3 is based on absolute continuity along almost every line parallel to every coordinate axes; at that time, the theory of absolutely continuous functions on the line was well understood. The goal of these lectures is to define W , H, and BL-spaces in the more general context of metric measure spaces and to show their equivalence. As we already wrote, this topic is also extensively covered in many research papers and monographs (see for instance [36, 37, 38] and the recent book [40]), covering in details also many more aspects of the theory 1 n Given an even, smooth, nonnegative convolution kernel ρ in R with compact support we denote by −n ρ"(x) = " ρ(x=") the sequence of rescaled kernels. 2 Motivations 3 (potential estimates, regularity of solutions, trace theorems, etc.). We want to convey a new point of view in this subject and show that the equivalence between these points of view and of the corresponding notions of gradient is a general phenomenon, true under almost no assumption on the metric measure structure. In addition, some concepts presented in these notes (even though with a few details, since we mostly focus on the Sobolev theory) are new even from the point of view of the classical theory, as the notion of \measure upper gradient" developed in connection with the theory of functions of bounded variation. Let us remark that, already in the Euclidean case, the three definitions are conceptually quite different. Indeed the three constructions rely on different objects: Definition 1.1 is based on coordinate vector fields, Definition 1.2 exploits approximations by smooth func- tions, whereas Definition 1.3 takes account of the behavior of a function along special curves. Moreover, the BL-space is characterized by a pointwise definition: there is no evidence there of Lebesgue equivalence classes of functions, as in the definition of the W -space. This is essentially due to the fact that at the time of Beppo Levi's work, the mathematical tools needed to deal with this kind of space were not completely developed yet. As a drawback, it was not clear whether the BL-space depended upon the choice of coordinate systems. It was much later that this criticism was overcome in Fuglede's paper [31], with a frame invariant definition that we will illustrate. 2. Motivations Most of the material in these lecture notes has been developed in a series of papers by Ambrosio, Gigli and Savar´e[11, 12, 14, 13, 15], as well as Ambrosio, Colombo, Di Marino [5]. In this series of papers, one of the main motivations for revisiting the theory of Sobolev spaces in metric measure spaces is that this theory provides basic mathematical tools for the theory of synthetic Ricci lower bounds in metric measure space. Let us describe roughly the two fundamental approaches in this context. One side of the theory was developed by Bakry and Emery´ who introduced Γ-calculus and used the language of Dirichlet forms to give a meaning to the inequality 1 1 ∆ jrfj2 − hrf; ∆rfi ≥ (∆f)2 + Kjrfj2; 2 N which is known as CD(K; N).