《不等式》本书旨在介绍大量运用于线性分析中的不等式，并且详细介绍它们的具体应用。本书以柯西不等式开头，grothendieck不等式结束，中间用许多不等式串成一个完整的篇幅，如，loomiswhitney不等式、最大值不等式、hardy 和 hilbert不等式、超收缩和拉格朗日索伯列夫不等、beckner以及等等。这些不等式可以用来研究函数空间的性质，它们之间的线性算子，以及绝对和算子。书中拥有许多完整和标准的结果，提供了许多应用，如勒贝格分解定理和勒贝格密度定理、希尔伯特变换和其他奇异积分算子、鞅收敛定理、特征值分布、lidskii积公式、mercer定理和littlewood 4/3定理。本书由(英)加林著。

Introduction

1 Measure and integral

 1.1 Measure

 1.2 Measurable functions

 1.3 Integration

 1.4 Notes and remarks

2 The Cauchy-Schwarz inequality

 2.1 Cauchy's inequality

 2.2 Inner-product spaces

 2.3 The Cauchy-Schwarz inequality

 2.4 Notes and remarks

3 The AM-GM inequality

 3.1 The AM-GM inequality

 3.2 Applications

 3.3 Notes and remarks

4 Convexity and Jensen's inequality

 4.1 Convex sets and convex functions

 4.2 Convex functions on an interval

 4.3 Directional derivatives and sublinear functionals

 4.4 The Hahn-Banach theorem

 4.5 Normed spaces, Banach spaces and Hilbert space

 4.6 The Hahn-Banach theorem for normed spaces

 4.7 Barycentres and weak integrals

 4.8 Notes and remarks

5 The Lp spaces

 5.1 Lp spaces, and Minkowski's inequality

 5.2 The Lebesgue decomposition theorem

 5.3 The reverse Minkowski inequality

 5.4 HSlder's inequality

 5.5 The inequalities of Liapounov and Littlewood

 5.6 Duality

 5.7 The Loomis-Whitney inequali'ty

 5.8 A Sobolev inequality

 5.9 Schur's theorem and Schur's test

 5.10 Hilbert's absolute inequality

 5.11 Notes and remarks

6 Banach function spaces

 6.1 Banach function spaces

 6.2 Function space duality

 6.3 Orlicz space

 6.4 Notes and remarks

7 Rearrangements

 7.1 Decreasing rearrangements

 7.2 Rearrangement-invariant Banach function spaces

 7.3 Muirhead's maximal function

 7.4 Majorization

 7.5 Calder6n's interpolation theorem and its converse

 7.6 Symmetric Banach sequence spaces

 7.7 The method of transference

 7.8 Finite doubly stochastic matrices

 7.9 Schur convexity

 7.10 Notes and remarks Maximal inequalities

 8.1 The Hardy-Riesz inequality

 8.2 The Hardy-Riesz inequality

 8.3 Related inequalities

 8.4 Strong type and weak type

 8.5 Riesz weak type

 8.6 Hardy, Littlewood, and a batsman's averages

 8.7 Riesz's sunrise lemma

 8.8 Differentiation almost everywhere

 8.9 Maximal operators in higher dimensions

 8.10 The Lebesgue density theorem

 8.11 Convolution kernels

 8.12 Hedberg's inequality

 8.13 Martingales

 8.14 Doob's inequality

 8.15 The martingale convergence theorem

 8.16 Notes and remarks

9 Complex interpolation

 9.1 Hadamard's three lines inequality

 9.2 Compatible couples and intermediate spaces

 9.3 The Riesz-Thorin interpolation theorem

 9.4 Young's inequality

 9.5 The Hausdorff-Young inequality

 9.6 Fourier type

 9.7 The generalized Clarkson inequalities

 9.8 Uniform convexity

 9.9 Notes and remarks

10 Real interpolation

 10.1 The Marcinkiewicz interpolation theorem: I

 10.2 Lorentz spaces

 10.3 Hardy's inequality

 10.4 The scale of Lorentz spaces

 10.5 The Marcinkiewicz interpolation theorem: II

 10.6 Notes and remarks

11 The Hilbert transform, and Hilbert's inequalities

 11.1 The conjugate Poisson kernel

 11.2 The Hilbert transform on

 11.3 The Hilbert transform on

 11.4 Hilbert's inequality for sequences

 11.5 The Hilbert transform on T

 11.6 Multipliers

 11.7 Singular integral operators

 11.8 Singular integral operators on

 11.9 Notes and remarks

12 Khintchine's inequality

 12.1 The contraction principle

 12.2 The reflection principle, and Lavy's inequalities

 12.3 Khintchine's inequality

 12.4 The law of the iterated logarithm

 12.5 Strongly embedded subspaces

 12.6 Stable random variables

 12.7 Sub-Gaussian random variables

 12.8 Kahane's theorem and Kahane's inequality

 12.9 Notes and remarks

13 Hypercontractive and logarithmic Sobolev inequalities

 13.1 Bonami's inequality

 13.2 Kahane's inequality revisited

 13.3 The theorem of Lataa and Oleszkiewicz

 13.4 The logarithmic Sobolev inequality on Dd

 13.5 Gaussian measure and the Hermite polynomials

 13.6 The central limit theorem

 13.7 The Gaussian hypercontractive inequality

 13.8 Correlated Gaussian random variables

 13.9 The Gaussian logarithmic Sobolev inequality

 13.10 The logarithmic Sobolev inequality in higher dimensions

 13.11 Beckner's inequality

 13.12 The Babenko-Beckner inequality

 13.13 Notes and remarks

14 Hadamard's inequality

 14.1 Hadamard's inequality

 14.2 Hadamard numbers

 14.3 Error-correcting codes

 14.4 Note and remark

15 Hilbert space operator inequalities

 15.1 Jordan normal form

 15.2 Riesz operators

 15.3 Related operators

 15.4 Compact operators

 15.5 Positive compact operators

 15.6 Compact operators between Hilbert spaces

 15.7 Singular numbers, and the Rayleigh-Ritz minimax formula

 15.8 Weyl's inequality and Horn's inequality

 15.9 Ky Fan's inequality

 15.10 Operator ideals

 15.11 The Hilbert-Schmidt class

 15.12 The trace class

 15.13 Lidskii's trace formula

 15.14 Operator ideal duality

 15.15 Notes and remarks

16 Summing operators

 16.1 Unconditional convergence

 16.2 Absolutely summing operators

 16.3 (p, q)-summing operators

 16.4 Examples of p-summing operators

 16.5 (p, 2)-summing operators between Hilbert spaces

 16.6 Positive operators on

 16.7 Mercer's theorem

 16.8 p-summing operators between Hilbert spaces

 16.9 Pietsch's domination theorem

 16.10 Pietsch's factorization theorem

 16.11 p-summing operators between Hilbert spaces

 16.12 The Dvoretzky-Rogers theorem

 16.13 Operators that factor through a Hilbert space

 16.14 Notes and remarks

17 Approximation numbers and eigenvalues

 17.1 The approximation, Gelfand and Weyl numbers

 17.2 Subadditive and submultiplicative properties

 17.3 Pietsch's inequality

 17.4 Eigenvalues of p-summing and (p, 2)-summing endomorphisms

 17.5 Notes and remarks

18 Grothendieck's inequality, type and cotype

 18.1 Littlewood's 4/3 inequality

 18.2 Grothendieck's inequality

 18.3 Grothendieck's theorem

 18.4 Another proof, using Paley's inequality

 18.5 The little Grothendieck theorem

 18.6 Type and cotype

 18.7 Gaussian type and cotype

 18.8 Type and cotype of LP spaces

 18.9 The little Grothendieck theorem revisited

 18.10 More on cotype

 18.11 Notes and remarks

References

Index of inequalities

Index