Analysis

by Elliott H. Lieb, Michael Loss

Overview

This is an excellent textbook on analysis and it has several unique features: Proofs of 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 $L^p$ 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, $L^p$ 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 (Schrodinger's 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

Product Details

ISBN-13:	9780821827833
Publisher:	American Mathematical Society
Publication date:	08/28/2001
Series:	Graduate Studies in Mathematics Series , #14
Edition description:	2ND
Pages:	348
Product dimensions:	7.20(w) x 10.20(h) x 1.10(d)

Table of Contents

	Preface to the First Edition	xvii
	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	11
1.5	Definition of measurable functions and integrals	12
1.6	Monotone convergence	17
1.7	Fatou'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
1.19	Approximation by C[superscript infinity] functions	36
	Exercises	37
Chapter 2.	L[superscript p]-Spaces	41
2.1	Definition of L[superscript p]-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 L[superscript p]-spaces	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 L[superscript p]([Omega])	61
2.15	Convolution	64
2.16	Approximation by C[superscript infinity]-functions	64
2.17	Separability of L[superscript p](R[superscript n])	67
2.18	Bounded sequences have weak limits	68
2.19	Approximation by C[superscript infinity subscript c]-functions	69
2.20	Convolution of functions in dual L[superscript p](R[superscript n])-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	110
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	123
5.1	Definition of the L[superscript 1] Fourier transform	123
5.2	Fourier transform of a Gaussian	125
5.3	Plancherel's theorem	126
5.4	Definition of the L[superscript 2] Fourier transform	127
5.5	Inversion formula	128
5.6	The Fourier transform in L[superscript p](R[superscript n])	128
5.7	The sharp Hausdorff-Young inequality	129
5.8	Convolutions	130
5.9	Fourier transform of |x|[superscript [alpha]-n]	130
5.10	Extension of 5.9 to L[superscript p](R[superscript n])	131
	Exercises	133
Chapter 6.	Distributions	135
6.1	Introduction	135
6.2	Test functions (The space D([Omega]))	136
6.3	Definition of distributions and their convergence	136
6.4	Locally summable functions, L[superscript p subscript loc]([Omega])	137
6.5	Functions are uniquely determined by distributions	138
6.6	Derivatives of distributions	139
6.7	Definition of W[superscript 1,p subscript loc]([Omega]) and W[superscript 1,p]([Omega])	140
6.8	Interchanging convolutions with distributions	142
6.9	Fundamental theorem of calculus for distributions	143
6.10	Equivalence of classical and distributional derivatives	144
6.11	Distributions with zero derivatives are constants	146
6.12	Multiplication and convolution of distributions by C[superscript infinity]-functions	146
6.13	Approximation of distributions by C[superscript infinity]-functions	147
6.14	Linear dependence of distributions	148
6.15	C[superscript infinity]([Omega]) is 'dense' in W[superscript 1,p subscript loc]([Omega])	149
6.16	Chain rule	150
6.17	Derivative of the absolute value	152
6.18	Min and Max of W[superscript 1,p]-functions are in W[superscript 1,p]	153
6.19	Gradients vanish on the inverse of small sets	154
6.20	Distributional Laplacian of Green's functions	156
6.21	Solution of Poisson's equation	157
6.22	Positive distributions are measures	159
6.23	Yukawa potential	163
6.24	The dual of W[superscript 1,p](R[superscript n])	166
	Exercises	167
Chapter 7.	The Sobolev Spaces H[superscript 1] and H[superscript 1/2]	171
7.1	Introduction	171
7.2	Definition of H[superscript 1]([Omega])	171
7.3	Completeness of H[superscript 1]([Omega])	172
7.4	Multiplication by functions in C[superscript infinity]([Omega])	173
7.5	Remark about H[superscript 1]([Omega]) and W[superscript 1,2]([Omega])	174
7.6	Density of C[superscript infinity]([Omega]) in H[superscript 1]([Omega])	174
7.7	Partial integration for functions in H[superscript 1](R[superscript n])	175
7.8	Convexity inequality for gradients	177
7.9	Fourier characterization of H[superscript 1](R[superscript n])	179
	Heat kernel	180
7.10	-[Delta] is the infinitesimal generator of the heat kernel	181
7.11	Definition of H[superscript 1/2](R[superscript n])	181
7.12	Integral formulas for (f,	184
7.13	Convexity inequality for the relativistic kinetic energy	185
7.14	Density of C[superscript infinity subscript c](R[superscript n]) in H[superscript 1/2](R[superscript n])	186
7.15	Action of [spuare root]-[Delta] and [spuare root]-[Delta] + m[superscript 2] - m on distributions	186
7.16	Multiplication of H[superscript 1/2] functions by C[superscript infinity]-functions	187
7.17	Symmetric decreasing rearrangement decreases kinetic energy	188
7.18	Weak limits	190
7.19	Magnetic fields: The H[superscript 1 subscript A]-spaces	191
7.20	Definition of H[superscript 1 subscript A](R[superscript n])	192
7.21	Diamagnetic inequality	193
7.22	C[superscript infinity subscript c](R[superscript n]) is dense in H[superscript 1 subscript A](R[superscript n])	194
	Exercises	195
Chapter 8.	Sobolev Inequalities	199
8.1	Introduction	199
8.2	Definition of D[superscript 1](R[superscript n]) and D[superscript 1/2](R[superscript n])	201
8.3	Sobolev's inequality for gradients	202
8.4	Sobolev's inequality for	|204
8.5	Sobolev inequalities in 1 and 2 dimensions	205
8.6	Weak convergence implies strong convergence on small sets	208
8.7	Weak convergence implies a.e. convergence	212
8.8	Sobolev inequalities for W[superscript m,p]([Omega])	213
8.9	Rellich-Kondrashov theorem	214
8.10	Nonzero weak convergence after translations	215
8.11	Poincare's inequalities for W[superscript m,p]([Omega])	218
8.12	Poincare-Sobolev inequality for W[superscript m,p]([Omega])	219
8.13	Nash's inequality	220
8.14	The logarithmic Sobolev inequality	223
8.15	A glance at contraction semigroups	225
8.16	Equivalence of Nash's inequality and smoothing estimates	227
8.17	Application to the heat equation	229
8.18	Derivation of the heat kernel via logarithmic Sobolev inequalities	232
	Exercises	235
Chapter 9.	Potential Theory and Coulomb Energies	237
9.1	Introduction	237
9.2	Definition of harmonic, subharmonic, and superharmonic functions	238
9.3	Properties of harmonic, subharmonic, and superharmonic functions	239
9.4	The strong maximum principle	243
9.5	Harnack's inequality	245
9.6	Subharmonic functions are potentials	246
9.7	Spherical charge distributions are 'equivalent' to point charges	248
9.8	Positivity properties of the Coulomb energy	250
9.9	Mean value inequality for [Delta] - [mu superscript 2]	251
9.10	Lower bounds on Schrodinger 'wave' functions	254
9.11	Unique solution of Yukawa's equation	255
	Exercises	256
Chapter 10.	Regularity of Solutions of Poisson's Equation	257
10.1	Introduction	257
10.2	Continuity and first differentiability of solutions of Poisson's equation	260
10.3	Higher differentiability of solutions of Poisson's equation	262
Chapter 11.	Introduction to the Calculus of Variations	267
11.1	Introduction	267
11.2	Schrodinger's equation	269
11.3	Domination of the potential energy by the kinetic energy	270
11.4	Weak continuity of the potential energy	274
11.5	Existence of a minimizer for E[subscript 0]	275
11.6	Higher eigenvalues and eigenfunctions	278
11.7	Regularity of solutions	279
11.8	Uniqueness of minimizers	280
11.9	Uniqueness of positive solutions	281
11.10	The hydrogen atom	282
11.11	The Thomas-Fermi problem	284
11.12	Existence of an unconstrained Thomas-Fermi minimizer	285
11.13	Thomas-Fermi equation	286
11.14	The Thomas-Fermi minimizer	287
11.15	The capacitor problem	289
11.16	Solution of the capacitor problem	293
11.17	Balls have smallest capacity	296
	Exercises	297
Chapter 12.	More about Eigenvalues	299
12.1	Min-max principles	300
12.2	Generalized min-max	302
12.3	Bound for eigenvalue sums in a domain	304
12.4	Bound for Schrodinger eigenvalue sums	306
12.5	Kinetic energy with antisymmetry	311
12.6	The semiclassical approximation	314
12.7	Definition of coherent states	316
12.8	Resolution of the identity	317
12.9	Representation of the nonrelativistic kinetic energy	319
12.10	Bounds for the relativistic kinetic energy	319
12.11	Large N eigenvalue sums in a domain	320
12.12	Large N asymptotics of Schrodinger eigenvalue sums	323
	Exercises	327
	List of Symbols	331
	References	335
	Index	341

What People are Saying About This

Steven G. Krantz

[The text was used as] a supplemental resource ... appropriate because Lieb and Loss's approach to this subject is quite unusual ... When looking at a particularly complicated set of ideas, I suggest [that students], "Have a look at Lieb/Loss. They have a nice way of cutting through all the difficulties and giving you a real picture of what is going on" ... I would certainly recommend this book [as a supplemental text] to colleagues who are teaching graduate real analysis." -- (Steven G. Krantz, Washington University, St. Louis)

Bruno Nachtergaele

Analysis is a unique book. It is written very much from the perspective of a user of analysis ... I do not know of any other book that shows so well what it takes to do research in applied and variational analysis or to solve real analysis problems as they naturally arise in mathematical physics and quantum mechanics ... The book is also very attractively produced ... The AMS is setting a new standard ... proving that quality editions do not have to be expensive ... [Students] found the concrete and constructive nature of the approach in the book very attractive." -- (Bruno Nachtergaele, University of California, Davis)

Lawrence E. Thomas

The book was ideal for my purposes, which included some introduction to some problems in mathematical physics, some `reinforcement' of what they already learned in real variable, and a good overview of the inequalities that working analysts need to use ... I personally find the book an extremely useful and convenient reference for research purposes. -- (Lawrence E. Thomas, University of Virginia, Charlottesville)

Editorial Reviews

Serves as a course in real analysis that begins with measure theory but also introduces more sophisticated material, including recent research. Topics include Lp-spaces, rearrangement inequalities, sharp integral inequalities, distribution theory, Fourier analysis, potential theory, and Sobolev spaces. Requires familiarity with calculus. Annotation c. by Book News, Inc., Portland, Or.

Booknews

Lieb and Loss offer a practical presentation of real and functional analysis at the beginning graduate level ... could be used as a two-semester introduction to graduate analysis ... not all of the topics covered are typical. The authors introduce the subject with a thorough presentation ... [an] informative exposition.

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