.

0 (hodnocen0 x )

BK

Rev. 2. ed.

Providence : American Mathematical Society, 2010

xxii, 348 s. : il. ; 26 cm

ISBN 978-0-8218-2783-3 (váz.)

Graduate studies in mathematics ; vol. 14

Obsahuje bibliografické odkazy a rejstřík

000195588

heat kernel estimates, the Nash inequality and the logarithmic Sobolev inequality are topics that are seldom treated on the level of a textbook. Best constants in several inequalities, such as Young’s inequality and the logarithmic Sobolev inequality, are also included. A thorough treatment of rearrangement inequalities and competing symmetries appears in book form for the first time. There is an extensive treatment of potential theory and its applications to quantum mechanics, which, again, is unique at this level. Uniform convexity of Lp space is treated very carefully. The presentation of this important subject is highly unusual for a textbook. All the proofs provide deep insights into the theorems. This book sets a new standard for a graduate textbook in analysis. // —Shing-Tung Yau, Harvard University // For some number of years, Rudin’s “Real and Complex", and a few other analysis books, served as the canonical choice for the book to use, and to teach from, in a first year grad analysis course. Lieb-Loss offers a refreshing alternative: It begins with a down-to-earth intro to measure theory, Lp and all that ...It aims at a wide range of essential applications, such as the Fourier transform, and series, inequalities, distributions, and Sobolev spaces—PDE, potential theory, calculus of variations, and math physics (Schrödingers equation, the hydrogen atom,Thomas-Fermi theory ...to mention a few). The book should work equally well in a one-, or in a two-semester course.

The first half of the book covers the basics, and the rest will be great for students to have, regardless of whether or not it gets to be included in a course. // —Palle E. T. Jorgensen, University of Iowa // ISBN 978-0-8218-2783-3 // For additional information ?? and updates on this book, visit ?? // www.ams.org/bookpages/gsm-14 // Preface to the First Edition // xvn // Preface to the Second Edition xxi // CHAPTER 1. Measure and Integration 1 // 1.1 Introduction 1 // 1.2 Basic notions of measure theory 4 // 1.3 Monotone class theorem 9 // 1.4 Uniqueness of measures ? // 1.5 Definition of measurable functions and integrals 12 // 1.6 Monotone convergence 17 // 1.7 Patou’s lemma 18 // 1.8 Dominated convergence 19 // 1.9 Missing term in Fatou’s lemma 21 // 1.10 Product measure 23 // 1.11 Commutativity and associativity of product measures 24 // 1.12 Fubini’s theorem 25 // 1.13 Layer cake representation 26 // 1.14 Bathtub principle 28 // 1.15 Constructing a measure from an outer measure 29 // 1.16 Uniform convergence except on small sets 31 // 1.17 Simple functions and really simple functions 32 // 1.18 Approximation by really simple functions 34 // IX // 1.19 Approximation by C00 functions // 36 // Exercises 37 // CHAPTER 2. Lp-Spaces 41 // 2.1 Definition of Lp-spaces 41 // 2.2 Jensen’s inequality 44 // 2.3 Holder’s inequality 45 // 2.4 Minkowski’s inequality 47 // 2.5 Hanner’s inequality 49 // 2.6 Differentiability of norms 51 // 2.7 Completeness of ZAspaces 52

2.8 Projection on convex sets 53 // 2.9 Continuous linear functionals and weak convergence 54 // 2.10 Linear functionals separate 56 // 2.11 Lower semicontinuity of norms 57 // 2.12 Uniform boundedness principle 58 // 2.13 Strongly convergent convex combinations 60 // 2.14 The dual of Il l) 61 // 2.15 Convolution 64 // 2.16 Approximation by C -functions 64 // 2.17 Separability of Lp(Mn) 67 // 2.18 Bounded sequences have weak limits 68 // 2.19 Approximation by C -functions 69 // 2.20 Convolutions of functions in dual Lp(Rn)-spaces are // continuous 70 // 2.21 Hilbert-spaces 71 // Exercises 75 // CHAPTER 3. Rearrangement Inequalities 79 // 3.1 Introduction 79 // 3.2 Definition of functions vanishing at infinity 80 // 3.3 Rearrangements of sets and functions 80 // 3.4 The simplest rearrangement inequality 82 // 3.5 Nonexpansivity of rearrangement 83 // 3.6 Riesz’s rearrangement inequality in one-dimension 84 // 3.7 Riesz’s rearrangement inequality 87 // 3.8 General rearrangement inequality 93 // 3.9 Strict rearrangement inequality 93 // Exercises 95 // CHAPTER 4. Integral Inequalities 97 // 4.1 Introduction 97 // 4.2 Young’s inequality 98 // 4.3 Hardy-Littlewood-Sobolev inequality 106 // 4.4 Conformal transformations and stereographic projection HO // 4.5 Conformal invariance of the Hardy-Littlewood-Sobolev // inequality 114 // 4.6 Competing symmetries 117 // 4.7 Proof of Theorem 4.3: Sharp version of the Hardy- // Littlewood-Sobolev inequality 119 // 4.8 Action of the conformal

group on optimizers 120 // Exercises 121 // CHAPTER 5. The Fourier Transform 125 // 5.1 Definition of the L1 Fourier transform 125 // 5.2 Fourier transform of a Gaussian 127 // 5.3 Plancherel’s theorem 128 // 5.4 Definition of the L2 Fourier transform 129 // 5.5 Inversion formula 130 // 5.6 The Fourier transform in Lp(Rn) 130 // 5.7 The sharp Hausdorff-Young inequality 131 // 5.8 Convolutions 132 // 5.9 Fourier transform of |x|Q_n 132 // 5.10 Extension of 5.9 to Lp(Mn) 133 // Exercises 133 // CHAPTER 6. Distributions 137 // 6.1 Introduction 137 // 6.2 Test functions (The space T>(Ł})) 138 // 6.3 Definition of distributions and their convergence 138 // 6.4 Locally summable functions, L oc(ii) 139 // 6.5 Functions are uniquely determined by distributions 140 // 6.6 Derivatives of distributions 141 // 6.7 Definition of (Q) and WliP(Q) 142 // 6.8 Interchanging convolutions with distributions 144 // 6.9 Fundamental theorem of calculus for distributions 145 // 6.10 Equivalence of classical and distributional derivatives 146 // 6.11 Distributions with zero derivatives are constants 148 // 6.12 Multiplication and convolution of distributions by (7°°- // functions 148 // 6.13 Approximation of distributions by C -functions 149 // 6.14 Linear dependence of distributions 150 // 6.15 ?°°(?) is ‘dense’ in 151 // 6.16 Chain rule 152 // 6.17 Derivative of the absolute value 154 // 6.18 Min and Max of lT1,p-functions are in W1,p 155 // 6.19 Gradients vanish on the inverse of small sets

157 // 6.20 Distributional Laplacian of Green’s functions 158 // 6.21 Solution of Poisson’s equation 159 // 6.22 Positive distributions are measures 161 // 6.23 Yukawa potential 166 // 6.24 The dual of TT1’ ) 169 // Exercises 170 // CHAPTER 7. The Sobolev Spaces H1 and H1/2 173 // 7.1 Introduction 173 // 7.2 Definition of ?1 ) 173 // 7.3 Completeness of ?1(?2) 174 // 7.4 Multiplication by functions in C00 ) 175 // 7.5 Remark about and VF1,2(f2) 176 // 7.6 Density of C00 ) in ?1(?) 176 // 7.7 Partial integration for functions in H1(Wl) 177 // 7.8 Convexity inequality for gradients 179 // Contents // Xlll // 7.9 Fourier characterization of ?1 (M71) 181 // • Heat kernel 182 // 7.10 — ? is the infinitesimal generator of the heat kernel 183 // 7.11 Definition of ?1/2(??) 183 // 7.12 Integral formulas for (/, \\p\\f) and (/, y/p2 +m2 f) 186 // 7.13 Convexity inequality for the relativistic kinetic // energy 187 // 7.14 Density of C {Rn) in ?1/2(??) 188 // 7.15 Action of ?/-? and ?/-? + m2 - m on distributions 188 // 7.16 Multiplication of ? -??? ??? by C -functions 189 // 7.17 Symmetric decreasing rearrangement decreases kinetic // energy 190 // 7.18 Weak limits 193 // 7.19 Magnetic fields: The ? -spaces 194 // 7.20 Definition of H\\(Rn) 194 // 7.21 Diamagnetic inequality 195 // 7.22 q?°(Mn) is dense in H\\(Rn) 196 // Exercises 197 // CHAPTER 8. Sobolev Inequalities 201 // 8.1 Introduction 201 // 8.2 Definition of D1(Rn) and ?1/2(??) 203 // 8.3 Sobolev’s inequality for

gradients 204 // 8.4 Sobolev’s inequality for \\p\\ 206 // 8.5 Sobolev inequalities in 1 and 2 dimensions 207 // 8.6 Weak convergence implies strong convergence on small sets 210 // 8.7 Weak convergence implies a.e. convergence 214 // 8.8 Sobolev inequalities for 215 // 8.9 Rellich-Kondrashov theorem 216 // 8.10 Nonzero weak convergence after translations 217 // 8.11 Poincaré’s inequalities for VFm,p(fl) 220 // 8.12 Poincaré-Sobolev inequality for 221 // 8.13 Nash’s inequality 222 // 8.14 The logarithmic Sobolev inequality 225 // 8.15 A glance at contraction semigroups 227 // 8.16 Equivalence of Nash’s inequality and smoothing estimates 229 // 8.17 Application to the heat equation 231 // 8.18 Derivation of the heat kernel via logarithmic Sobolev inequalities 234 // Exercises 237 // CHAPTER 9. Potential Theory and Coulomb Energies 239 // 9.1 Introduction 239 // 9.2 Definition of harmonic, subharmonic, and superharmonic // functions 240 // 9.3 Properties of harmonic, subharmonic, and superharmonic // functions 241 // 9.4 The strong maximum principle 246 // 9.5 Harnack’s inequality 247 // 9.6 Subharmonic functions are potentials 248 // 9.7 Spherical charge distributions are ‘equivalent’ to point // charges 251 // 9.8 Positivity properties of the Coulomb energy 252 // 9.9 Mean value inequality for ? — /i2 254 // 9.10 Lower bounds on Schrödinger ‘wave’ functions 256 // 9.11 Unique solution of Yukawa’s equation 257 // Exercises 258 // CHAPTER 10. Regularity

of Solutions of Poisson’s // Equation 259 // 10.1 Introduction 259 // 10.2 Continuity and first differentiability of solutions of Poisson’s // equation 262 // 10.3 Higher differentiability of solutions of Poisson’s equation 264 // CHAPTER 11. Introduction to the Calculus of Variations 269 // 11.1 Introduction 269 // 11.2 Schrödinger’s equation 271 // 11.3 Domination of the potential energy by the kinetic energy 272 // 11.4 Weak continuity of the potential energy 276 // 11.5 Existence of a minimizer for Eq 277 // 11.6 Higher eigenvalues and eigenfunctions 279 // 11.7 Regularity of solutions 281 // 11.8 Uniqueness of minimizers 282 // 11.9 Uniqueness of positive solutions 283 // 11.10 The hydrogen atom 284 // 11.11 The Thomas-Fermi problem 285 // 11.12 Existence of an unconstrained Thomas-Fermi minimizer 286 // 11.13 Thomas-Fermi equation 287 // 11.14 The Thomas-Fermi minimizer 289 // 11.15 The capacitor problem 291 // 11.16 Solution of the capacitor problem 295 // 11.17 Balls have smallest capacity 298 // Exercises 299 // CHAPTER 12. More about Eigenvalues 301 // 12.1 Min-max principles 302 // 12.2 Generalized min-max 304 // 12.3 Bound for eigenvalue sums in a domain 306 // 12.4 Bound for Schrödinger eigenvalue sums 308 // 12.5 Kinetic energy with antisymmetry 313 // 12.6 The semiclassical approximation 316 // 12.7 Definition of coherent states 318 // 12.8 Resolution of the identity 319 // 12.9 Representation of the nonrelativistic kinetic energy 321 // 12.10 Bounds for

the relativistic kinetic energy 321 // 12.11 Large N eigenvalue sums in a domain 322 // 12.12 Large N asymptotics of Schrödinger eigenvalue sums 325 // Exercises 329 // List of Symbols 333 // References 337 // Index // 343