Cover

Title

Copyright

Contents

Preface to the First Edition

Preface to the Second Edition

CHAPTER 1. Measure and Integration

1.1 Introduction

1.2 Basic notions of measure theory

1.3 Monotone class theorem

1.4 Uniqueness of measures

1.5 Definition of measurable functions and integrals

1.6 Monotone convergence

1.7 Fatou's lemma

1.8 Dominated convergence

1.9 Missing term in Fatou's lemma

1.10 Product measure

1.11 Commutativity and associativity of product measures

1.12 Fubini's theorem

1.13 Layer cake representation

1.14 Bathtub principle

1.15 Constructing a measure from an outer measure

1.16 Uniform convergence except on small sets

1.17 Simple functions and really simple functions

1.18 Approximation by really simple functions

1.19 Approximation by C∞ functions

Exercises

CHAPTER 2. L[sup(p)]-Spaces

2.1 Definition of L[sup(p)]-spaces

2.2 Jensen's inequality

2.3 Holder's inequality

2.4 Minkowski's inequality

2.5 Hanner's inequality

2.6 Differentiability of norms

2.7 Completeness of L[sup(p)]-spaces

2.8 Projection on convex sets

2.9 Continuous linear functionals and weak convergence

2.10 Linear functionals separate

2.11 Lower semicontinuity of norms

2.12 Uniform boundedness principle

2.13 Strongly convergent convex combinations

2.14 The dual of L[sup(p)](Ω)

2.15 Convolution

2.16 Approximation by C∞-functions

2.17 Separability of L[sup(p)](R[sup(n)])

2.18 Bounded sequences have weak limits

2.19 Approximation by C[sup(∞)][sub(c)]-functions

2.20 Convolutions of functions in dual L[sup(p)](R[sup(p)])-spaces are continuous

2.21 Hilbert-spaces

Exercises

CHAPTER 3. Rearrangement Inequalities

3.1 Introduction

3.2 Definition of functions vanishing at infinity

3.3 Rearrangements of sets and functions

3.4 The simplest rearrangement inequality

3.5 Nonexpansivity of rearrangement

3.6 Riesz's rearrangement inequality in one-dimension

3.7 Riesz's rearrangement inequality

3.8 General rearrangement inequality

3.9 Strict rearrangement inequality

Exercises

CHAPTER 4. Integral Inequalities

4.1 Introduction

4.2 Young's inequality

4.3 Hardy–Littlewood–Sobolev inequality

4.4 Conformal transformations and stereographic projection

4.5 Conformal invariance of the Hardy–Littlewood–Sobolev inequality

4.6 Competing symmetries

4.7 Proof of Theorem 4.3: Sharp version of the Hardy–Littlewood–Sobolev inequality

4.8 Action of the conformal group on optimizers

Exercises

CHAPTER 5. The Fourier Transform

5.1 Definition of the L[sup(1)] Fourier transform

5.2 Fourier transform of a Gaussian

5.3 Plancherels theorem

5.4 Definition of the L[sup(2)] Fourier transform

5.5 Inversion formula

5.6 The Fourier transform in L[sup(p)](R[sup(n)])

5.7 The sharp Hausdorff–Young inequality

5.8 Convolutions

5.9 Fourier transform of |x|[sup(α…n)]

5.10 Extension of 5.9 to L[sup(p)](R[sup(n)])

Exercises

CHAPTER 6. Distributions

6.1 Introduction

6.2 Test functions (The space D(Ω))

6.3 Definition of distributions and their convergence

6.4 Locally summable functions, L[sup(p)][sub(loc)](Ω)

6.5 Functions are uniquely determined by distributions

6.6 Derivatives of distributions

6.7 Definition of w[sup(1,p)][sub(loc)](Ω) and W[sup(1,p)](Ω)

6.8 Interchanging convolutions with distributions

6.9 Fundamental theorem of calculus for distributions

6.10 Equivalence of classical and distributional derivatives

6.11 Distributions with zero derivatives are constants

6.12 Multiplication and convolution of distributions by C[sup(∞)]-functions

6.13 Approximation of distributions by C[sup(∞)]-functions

6.14 Linear dependence of distributions

6.15 C[sup(∞)]([sup(937;)]) is 'dense' in W[sup(1,p)][sub(loc)](Ω)

6.16 Chain rule

6.17 Derivative of the absolute value

6.18 Min and Max of W[sup(1,p)]-functions are in W[sup(1,p)]

6.19 Gradients vanish on the inverse of small sets

6.20 Distributional Laplacian of Green's functions

6.21 Solution of Poisson's equation

6.22 Positive distributions are measures

6.23 Yukawa potential

6.24 The dual of W[sup(1,p)](R[sup(n)])

Exercises

CHAPTER 7. The Sobolev Spaces H[sup(1)] and H[sup(1/2)]

7.1 Introduction

7.2 Definition of H[sup(1)](Ω)

7.3 Completeness of H[sup(1)](Ω)

7.4 Multiplication by functions in C[sup(∞)](Ω)

7.5 Remark about H[sup(1)](Ω) and W[sup(1,2)](Ω)

7.6 Density of C∞(Ω) in H[sup(1)](Ω)

7.7 Partial integration for functions in H[sup(1)](R[sup(n)])

7.8 Convexity inequality for gradients

7.9 Fourier characterization of H[sup(1)](R[sup(n)])

Heat kernel

7.10 …Δ is the infinitesimal generator of the heat kernel

7.11 Definition of H[sup(1/2)](R[sup(n)])

7.12 Integral formulas for (f, |p|f) and (f, [omitted] f)

7.13 Convexity inequality for the relativistic kinetic energy

7.14 Density of C[sup(∞)][sub(c)](R[sup(n)]) in H[sup(1/2)](R[sup(n)])

7.15 Action of [omitted] and [omitted] – m on distributions

7.16 Multiplication of H[sup(1/2)]-functions by C∞-functions

7.17 Symmetric decreasing rearrangement decreases kinetic energy

7.18 Weak limits

7.19 Magnetic fields: The H[sup(1)][sub(A)]-spaces

7.20 Definition of H[sup(1)][sub(A)](R[sup(n)])

7.21 Diamagnetic inequality

7.22 C[sup(∞)][sup(c)](R[sup(n)]) is dense in H[sup(1)][sub(A)](R[sup(n)])

Exercises

CHAPTER 8. Sobolev Inequalities

8.1 Introduction

8.2 Definition of D[sup(1)](R[sup(n)]) and D[sup(1/2)](R[sup(n)])

8.3 Sobolev's inequality for gradients

8.4 Sobolev's inequality for |p|

8.5 Sobolev inequalities in 1 and 2 dimensions

8.6 Weak convergence implies strong convergence on small sets

8.7 Weak convergence implies a.e. convergence

8.8 Sobolev inequalities for W[sup(m,p)](Ω)

8.9 Rellich-Kondrashov theorem

8.10 Nonzero weak convergence after translations

8.11 Poincaré's inequalities for W[sup(m,p)](Ω)

8.12 Poincaré-Sobolev inequality for W[sup(m,p)](Ω)

8.13 Nash's inequality

8.14 The logarithmic Sobolev inequality

8.15 A glance at contraction semigroups

8.16 Equivalence of Nash's inequality and smoothing estimates

8.17 Application to the heat equation

8.18 Derivation of the heat kernel via logarithmic Sobolev in-equalities

Exercises

CHAPTER 9. Potential Theory and Coulomb Energies

9.1 Introduction

9.2 Definition of harmonic, subharmonic, and superharmonic functions

9.3 Properties of harmonic, subharmonic, and superharmonic functions

9.4 The strong maximum principle

9.5 Harnack's inequality

9.6 Subharmonic functions are potentials

9.7 Spherical charge distributions are ' equivalent' to point charges

9.8 Positivity properties of the Coulomb energy

9.9 Mean value inequality for Δ … μ[sup(2)]

9.10 Lower bounds on Schrödinger 'wave' functions

9.11 Unique solution of Yukawa's equation

Exercises

CHAPTER 10. Regularity of Solutions of Poisson's Equation

10.1 Introduction

10.2 Continuity and first differentiability of solutions of Poisson's equation

10.3 Higher differentiability of solutions of Poisson's equation

CHAPTER 11. Introduction to the Calculus of Variations

11.1 Introduction

11.2 Schrödinger's equation

11.3 Domination of the potential energy by the kinetic energy

11.4 Weak continuity of the potential energy

11.5 Existence of a minimizer for E[sup(0)]

11.6 Higher eigenvalues and eigenfunctions

11.7 Regularity of solutions

11.8 Uniqueness of minimizers

11.9 Uniqueness of positive solutions

11.10 The hydrogen atom

11.11 The Thomas–Fermi problem

11.12 Existence of an unconstrained Thomas–Fermi minimizer

11.13 Thomas–Fermi equation

11.14 The Thomas–Fermi minimizer

11.15 The capacitor problem

11.16 Solution of the capacitor problem

11.17 Balls have smallest capacity

Exercises

CHAPTER 12. More about Eigenvalues

12.1 Min-max principles

12.2 Generalized min-max

12.3 Bound for eigenvalue sums in a domain

12.4 Bound for Schrodinger eigenvalue sums

12.5 Kinetic energy with antisymmetry

12.6 The semiclassical approximation

12.7 Definition of coherent states

12.8 Resolution of the identity

12.9 Representation of the nonrelativistic kinetic energy

12.10 Bounds for the relativistic kinetic energy

12.11 Large N eigenvalue sums in a domain

12.12 Large N asymptotics of Schrodinger eigenvalue sums

Exercises

List of Symbols

References

Index

A

B

C

D

E

F

G

H

I

J

K

L

M

N

O

P

R

S

T

U

V

W

Y

Back Cover