Five Lectures on Optimal Transportation: Geometry, Regularity and Applications
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Sobolev Spaces, Theory and Applications
Sobolev spaces, theory and applications Piotr Haj lasz1 Introduction These are the notes that I prepared for the participants of the Summer School in Mathematics in Jyv¨askyl¨a,August, 1998. I thank Pekka Koskela for his kind invitation. This is the second summer course that I delivere in Finland. Last August I delivered a similar course entitled Sobolev spaces and calculus of variations in Helsinki. The subject was similar, so it was not posible to avoid overlapping. However, the overlapping is little. I estimate it as 25%. While preparing the notes I used partially the notes that I prepared for the previous course. Moreover Lectures 9 and 10 are based on the text of my joint work with Pekka Koskela [33]. The notes probably will not cover all the material presented during the course and at the some time not all the material written here will be presented during the School. This is however, not so bad: if some of the results presented on lectures will go beyond the notes, then there will be some reasons to listen the course and at the same time if some of the results will be explained in more details in notes, then it might be worth to look at them. The notes were prepared in hurry and so there are many bugs and they are not complete. Some of the sections and theorems are unfinished. At the end of the notes I enclosed some references together with comments. This section was also prepared in hurry and so probably many of the authors who contributed to the subject were not mentioned. -
Extremal Dependence Concepts
Extremal dependence concepts Giovanni Puccetti1 and Ruodu Wang2 1Department of Economics, Management and Quantitative Methods, University of Milano, 20122 Milano, Italy 2Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON N2L3G1, Canada Journal version published in Statistical Science, 2015, Vol. 30, No. 4, 485{517 Minor corrections made in May and June 2020 Abstract The probabilistic characterization of the relationship between two or more random variables calls for a notion of dependence. Dependence modeling leads to mathematical and statistical challenges and recent developments in extremal dependence concepts have drawn a lot of attention to probability and its applications in several disciplines. The aim of this paper is to review various concepts of extremal positive and negative dependence, including several recently established results, reconstruct their history, link them to probabilistic optimization problems, and provide a list of open questions in this area. While the concept of extremal positive dependence is agreed upon for random vectors of arbitrary dimensions, various notions of extremal negative dependence arise when more than two random variables are involved. We review existing popular concepts of extremal negative dependence given in literature and introduce a novel notion, which in a general sense includes the existing ones as particular cases. Even if much of the literature on dependence is focused on positive dependence, we show that negative dependence plays an equally important role in the solution of many optimization problems. While the most popular tool used nowadays to model dependence is that of a copula function, in this paper we use the equivalent concept of a set of rearrangements. -
Introduction to Sobolev Spaces
Introduction to Sobolev Spaces Lecture Notes MM692 2018-2 Joa Weber UNICAMP December 23, 2018 Contents 1 Introduction1 1.1 Notation and conventions......................2 2 Lp-spaces5 2.1 Borel and Lebesgue measure space on Rn .............5 2.2 Definition...............................8 2.3 Basic properties............................ 11 3 Convolution 13 3.1 Convolution of functions....................... 13 3.2 Convolution of equivalence classes................. 15 3.3 Local Mollification.......................... 16 3.3.1 Locally integrable functions................. 16 3.3.2 Continuous functions..................... 17 3.4 Applications.............................. 18 4 Sobolev spaces 19 4.1 Weak derivatives of locally integrable functions.......... 19 1 4.1.1 The mother of all Sobolev spaces Lloc ........... 19 4.1.2 Examples........................... 20 4.1.3 ACL characterization.................... 21 4.1.4 Weak and partial derivatives................ 22 4.1.5 Approximation characterization............... 23 4.1.6 Bounded weakly differentiable means Lipschitz...... 24 4.1.7 Leibniz or product rule................... 24 4.1.8 Chain rule and change of coordinates............ 25 4.1.9 Equivalence classes of locally integrable functions..... 27 4.2 Definition and basic properties................... 27 4.2.1 The Sobolev spaces W k;p .................. 27 4.2.2 Difference quotient characterization of W 1;p ........ 29 k;p 4.2.3 The compact support Sobolev spaces W0 ........ 30 k;p 4.2.4 The local Sobolev spaces Wloc ............... 30 4.2.5 How the spaces relate.................... 31 4.2.6 Basic properties { products and coordinate change.... 31 i ii CONTENTS 5 Approximation and extension 33 5.1 Approximation............................ 33 5.1.1 Local approximation { any domain............. 33 5.1.2 Global approximation on bounded domains....... -
The Geometry of Optimal Transportation
THE GEOMETRY OF OPTIMAL TRANSPORTATION Wilfrid Gangbo∗ School of Mathematics, Georgia Institute of Technology Atlanta, GA 30332, USA [email protected] Robert J. McCann∗ Department of Mathematics, Brown University Providence, RI 02912, USA and Institut des Hautes Etudes Scientifiques 91440 Bures-sur-Yvette, France [email protected] July 28, 1996 Abstract A classical problem of transporting mass due to Monge and Kantorovich is solved. Given measures µ and ν on Rd, we find the measure-preserving map y(x) between them with minimal cost | where cost is measured against h(x − y)withhstrictly convex, or a strictly concave function of |x − y|.This map is unique: it is characterized by the formula y(x)=x−(∇h)−1(∇ (x)) and geometrical restrictions on . Connections with mathematical economics, numerical computations, and the Monge-Amp`ere equation are sketched. ∗Both authors gratefully acknowledge the support provided by postdoctoral fellowships: WG at the Mathematical Sciences Research Institute, Berkeley, CA 94720, and RJM from the Natural Sciences and Engineering Research Council of Canada. c 1995 by the authors. To appear in Acta Mathematica 1997. 1 2 Key words and phrases: Monge-Kantorovich mass transport, optimal coupling, optimal map, volume-preserving maps with convex potentials, correlated random variables, rearrangement of vector fields, multivariate distributions, marginals, Lp Wasserstein, c-convex, semi-convex, infinite dimensional linear program, concave cost, resource allocation, degenerate elliptic Monge-Amp`ere 1991 Mathematics -
(Measure Theory for Dummies) UWEE Technical Report Number UWEETR-2006-0008
A Measure Theory Tutorial (Measure Theory for Dummies) Maya R. Gupta {gupta}@ee.washington.edu Dept of EE, University of Washington Seattle WA, 98195-2500 UWEE Technical Report Number UWEETR-2006-0008 May 2006 Department of Electrical Engineering University of Washington Box 352500 Seattle, Washington 98195-2500 PHN: (206) 543-2150 FAX: (206) 543-3842 URL: http://www.ee.washington.edu A Measure Theory Tutorial (Measure Theory for Dummies) Maya R. Gupta {gupta}@ee.washington.edu Dept of EE, University of Washington Seattle WA, 98195-2500 University of Washington, Dept. of EE, UWEETR-2006-0008 May 2006 Abstract This tutorial is an informal introduction to measure theory for people who are interested in reading papers that use measure theory. The tutorial assumes one has had at least a year of college-level calculus, some graduate level exposure to random processes, and familiarity with terms like “closed” and “open.” The focus is on the terms and ideas relevant to applied probability and information theory. There are no proofs and no exercises. Measure theory is a bit like grammar, many people communicate clearly without worrying about all the details, but the details do exist and for good reasons. There are a number of great texts that do measure theory justice. This is not one of them. Rather this is a hack way to get the basic ideas down so you can read through research papers and follow what’s going on. Hopefully, you’ll get curious and excited enough about the details to check out some of the references for a deeper understanding. -
L P and Sobolev Spaces
NOTES ON Lp AND SOBOLEV SPACES STEVE SHKOLLER 1. Lp spaces 1.1. Definitions and basic properties. Definition 1.1. Let 0 < p < 1 and let (X; M; µ) denote a measure space. If f : X ! R is a measurable function, then we define 1 Z p p kfkLp(X) := jfj dx and kfkL1(X) := ess supx2X jf(x)j : X Note that kfkLp(X) may take the value 1. Definition 1.2. The space Lp(X) is the set p L (X) = ff : X ! R j kfkLp(X) < 1g : The space Lp(X) satisfies the following vector space properties: (1) For each α 2 R, if f 2 Lp(X) then αf 2 Lp(X); (2) If f; g 2 Lp(X), then jf + gjp ≤ 2p−1(jfjp + jgjp) ; so that f + g 2 Lp(X). (3) The triangle inequality is valid if p ≥ 1. The most interesting cases are p = 1; 2; 1, while all of the Lp arise often in nonlinear estimates. Definition 1.3. The space lp, called \little Lp", will be useful when we introduce Sobolev spaces on the torus and the Fourier series. For 1 ≤ p < 1, we set ( 1 ) p 1 X p l = fxngn=1 j jxnj < 1 : n=1 1.2. Basic inequalities. Lemma 1.4. For λ 2 (0; 1), xλ ≤ (1 − λ) + λx. Proof. Set f(x) = (1 − λ) + λx − xλ; hence, f 0(x) = λ − λxλ−1 = 0 if and only if λ(1 − xλ−1) = 0 so that x = 1 is the critical point of f. In particular, the minimum occurs at x = 1 with value f(1) = 0 ≤ (1 − λ) + λx − xλ : Lemma 1.5. -
The Logarithmic Sobolev Inequality Along the Ricci Flow
The Logarithmic Sobolev Inequality Along The Ricci Flow (revised version) Rugang Ye Department of Mathematics University of California, Santa Barbara July 20, 2007 1. Introduction 2. The Sobolev inequality 3. The logarithmic Sobolev inequality on a Riemannian manifold 4. The logarithmic Sobolev inequality along the Ricci flow 5. The Sobolev inequality along the Ricci flow 6. The κ-noncollapsing estimate Appendix A. The logarithmic Sobolev inequalities on the euclidean space Appendix B. The estimate of e−tH Appendix C. From the estimate for e−tH to the Sobolev inequality 1 Introduction Consider a compact manifold M of dimension n 3. Let g = g(t) be a smooth arXiv:0707.2424v4 [math.DG] 29 Aug 2007 solution of the Ricci flow ≥ ∂g = 2Ric (1.1) ∂t − on M [0, T ) for some (finite or infinite) T > 0 with a given initial metric g(0) = g . × 0 Theorem A For each σ > 0 and each t [0, T ) there holds ∈ R n σ u2 ln u2dvol σ ( u 2 + u2)dvol ln σ + A (t + )+ A (1.2) ≤ |∇ | 4 − 2 1 4 2 ZM ZM 1 for all u W 1,2(M) with u2dvol =1, where ∈ M R 4 A1 = 2 min Rg0 , ˜ 2 n − CS(M,g0) volg0 (M) n A = n ln C˜ (M,g )+ (ln n 1), 2 S 0 2 − and all geometric quantities are associated with the metric g(t) (e.g. the volume form dvol and the scalar curvature R), except the scalar curvature Rg0 , the modified Sobolev ˜ constant CS(M,g0) (see Section 2 for its definition) and the volume volg0 (M) which are those of the initial metric g0. -
27. Sobolev Inequalities 27.1
ANALYSIS TOOLS WITH APPLICATIONS 493 27. Sobolev Inequalities 27.1. Morrey’s Inequality. d 1 d Notation 27.1. Let S − be the sphere of radius one centered at zero inside R . d 1 d For a set Γ S − ,x R , and r (0, ), let ⊂ ∈ ∈ ∞ Γx,r x + sω : ω Γ such that 0 s r . ≡ { ∈ ≤ ≤ } So Γx,r = x + Γ0,r where Γ0,r is a cone based on Γ, seeFigure49below. Γ Γ Figure 49. The cone Γ0,r. d 1 Notation 27.2. If Γ S − is a measurable set let Γ = σ(Γ) be the surface “area” of Γ. ⊂ | | Notation 27.3. If Ω Rd is a measurable set and f : Rd C is a measurable function let ⊂ → 1 fΩ := f(x)dx := f(x)dx. − m(Ω) Ω ZΩ Z By Theorem 8.35, r d 1 (27.1) f(y)dy = f(x + y)dy = dt t − f(x + tω) dσ(ω) Γx,r Γ0,r 0 Z Z Z ZΓ and letting f =1in this equation implies d (27.2) m(Γx,r)= Γ r /d. | | d 1 Lemma 27.4. Let Γ S − be a measurable set such that Γ > 0. For u 1 ⊂ | | ∈ C (Γx,r), 1 u(y) (27.3) u(y) u(x) dy |∇ d | 1 dy. − | − | ≤ Γ x y − ΓZx,r | |ΓZx,r | − | 494 BRUCE K. DRIVER† d 1 Proof. Write y = x + sω with ω S − , then by the fundamental theorem of calculus, ∈ s u(x + sω) u(x)= u(x + tω) ωdt − ∇ · Z0 and therefore, s u(x + sω) u(x) dσ(ω) u(x + tω) dσ(ω)dt | − | ≤ 0 Γ |∇ | ZΓ Z Z s d 1 u(x + tω) = t − dt |∇ d | 1 dσ(ω) 0 Γ x + tω x − Z Z | − | u(y) u(y) = |∇ d | 1 dy |∇ d | 1 dy, y x − ≤ x y − ΓZx,s | − | ΓZx,r | − | wherein the second equality we have used Eq. -
Sobolev Inequalities in Familiar and Unfamiliar Settings
Sobolev Inequalities in Familiar and Unfamiliar Settings Laurent Salo®-Coste Abstract The classical Sobolev inequalities play a key role in analysis in Euclidean spaces and in the study of solutions of partial di®erential equations. In fact, they are extremely flexible tools and are useful in many di®erent settings. This paper gives a glimpse of assortments of such applications in a variety of contexts. 1 Introduction There are few articles that have turned out to be as influential and truly im- portant as S.L. Sobolev 1938 article [93] (the American translation appeared in 1963), where he introduces his famed inequalities. It is the idea of a func- tional inequality itself that Sobolev brings to life in his paper, as well as the now so familiar notion of an a priori inequality, i.e., a functional inequality established under some strong hypothesis and that might be extended later, perhaps almost automatically, to its natural domain of de¯nition. (These ideas are also related to the theory of distributions which did not exist at the time and whose magni¯cent development by L. Schwartz was, in part, anticipated in the work of S.L. Sobolev.) The most basic and important applications of Sobolev inequalities are to the study of partial di®erential equations. Simply put, Sobolev inequalities provide some of the very basic tools in the study of the existence, regularity, and uniqueness of the solutions of all sorts of partial di®erential equations, lin- ear and nonlinear, elliptic, parabolic, and hyperbolic. I leave to others, much better quali¯ed than me, to discuss these beautiful developments. -
A Theory of Radon Measures on Locally Compact Spaces
A THEORY OF RADON MEASURES ON LOCALLY COMPACT SPACES. By R. E. EDWARDS. 1. Introduction. In functional analysis one is often presented with the following situation: a locally compact space X is given, and along with it a certain topological vector space • of real functions defined on X; it is of importance to know the form of the most general continuous linear functional on ~. In many important cases, s is a superspace of the vector space ~ of all reul, continuous functions on X which vanish outside compact subsets of X, and the topology of s is such that if a sequence (]n) tends uniformly to zero and the ]n collectively vanish outside a fixed compact subset of X, then (/~) is convergent to zero in the sense of E. In this case the restriction to ~ of any continuous linear functional # on s has the property that #(/~)~0 whenever the se- quence (/~) converges to zero in the manner just described. It is therefore an important advance to determine all the linear functionals on (~ which are continuous in this eense. It is customary in some circles (the Bourbaki group, for example) to term such a functional # on ~ a "Radon measure on X". Any such functional can be written in many ways as the difference of two similar functionals, euch having the additional property of being positive in the sense that they assign a number >_ 0 to any function / satisfying ] (x) > 0 for all x E X. These latter functionals are termed "positive Radon measures on X", and it is to these that we may confine our attention. -
Sharp Log–Sobolev Inequalities
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 10, October 1998, Pages 2903{2904 S 0002-9939(98)04406-2 SHARP LOG{SOBOLEV INEQUALITIES OSCAR S. ROTHAUS (Communicated by Palle E. T. Jorgensen) Abstract. We show existence of a wide variety of Log{Sobolev inequalities in which the constant is exactly that required by the Poincar´e inequality which may be inferred from the Log{Sobolev. We are given a smooth compact Riemannian manifold M, intrinsic gradient , and volume element dµ — we assume W.L.O.G. that µ(M)=1—andasmooth∇ positive function m, giving rise to a probability measure mdµ = dm, for which we have a log–Sobolev inequality (LSI): (1) ρ f 2dm f 2 ln f 2dm f 2dm ln f 2dm: |∇ | ≥ | | | | − | | | | ZM ZM ZM ZM It is well known that ρ 2/λ,whereλis the first non–zero eigenvalue of the Schr¨odinger operator ≥ m f ∆f + f ∇ ; → ∇ · m (∆ the usual Laplacian) attached to the Dirichlet form in (1). We call the LSI sharp if ρ =2/λ. Many examples of sharp inequalities are known, the most familiar arising from M the sphere with the usual metric scaled to give M unit volume, and m =1. We will show here that for every compact homogeneous Riemannian manifold, there are a continuum of choices of m for which sharp LSI’s exist. We follow the notation and conclusions of [1], which we now briefly review. For every M as described initially there is a least constant ρ0(M), the hypercontractive constant for M, such that (2) ρ (M) f 2dµ f 2 ln f 2dµ f 2dµ ln f 2dµ. -
Sobolev's Inequality Under a Curvature-Dimension Condition
Sobolev’s inequality under a curvature-dimension condition Louis Dupaigne, Ivan Gentil, Simon Zugmeyer To cite this version: Louis Dupaigne, Ivan Gentil, Simon Zugmeyer. Sobolev’s inequality under a curvature-dimension condition. 2021. hal-03006155v3 HAL Id: hal-03006155 https://hal.archives-ouvertes.fr/hal-03006155v3 Preprint submitted on 19 Jan 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Sobolev's inequality under a curvature-dimension condition Louis Dupaigne, Ivan Gentil, Simon Zugmeyer January 19, 2021 R´esum´e Dans cette note, nous proposons une nouvelle preuve de l'in´egalit´ede Sobolev sur les vari´et´es`a courbure de Ricci minor´eepar une constante positive. Le r´esultatavait ´et´eobtenu en 1983 par Ilias. Nous pr´esentons une preuve tr`escourte de ce th´eor`eme, dressons l'´etatde l'art pour cette fameuse in´egalit´eet expliquons en quoi notre m´ethode, qui repose sur un flot de gradient, est simple et robuste. En particulier, nous ´elucidonsles calculs utilis´esdans des travaux pr´ec´edents, `acommencer par un c´el`ebrearticle de Bidaut-V´eron et V´eronpubli´een 1991. Abstract In this note we present a new proof of Sobolev's inequality under a uniform lower bound of the Ricci curvature.