格拉法克斯编著的《现代傅里叶分析》的作品旨在为读者提供学习欧几里得调和解析领域的理论基础。原始版本是以单卷集发布的，但是由于其体积、范围和新材料的增加，第二版改为两卷集发行。目前的这个版本包括了新的一章讲述时频分析和Carleson-Hunt定理。第一卷包括一些基础经典话题，包括插值、傅里叶级数、傅里叶变换、极大值函数、奇异积分和Littlewood-Paley定理。第二卷包括更多现代话题，如函数空间、原子分解、非卷积型的奇异积分和权重不等式。

6 smoothness and function spaces

 6.1 riesz and bessel potentials, fractional integrals

6.1.1 riesz potentials

6.1.2 bessel potentials

exercises

 6.2 sobolev spaces

6.2.1 definition and basic properties of general sobolev spaces

6.2.2 littlewood-paley characterization of inhomogeneous

sobolev spaces

6.2.3 littlewood-paley characterization of homogeneous

sobolev spaces

exercises

 6.3 lipschitz spaces

6.3.1 introduction to lipschitz spaces

6.3.2 littlewood-paley characterization of homogeneous

lipschitz spaces

6.3.3 littlewood-paley characterization of inhomogeneous

lipschitz spaces

exercises

 6.4 hardy spaces

6.4.1 definition of hardy spaces

6.4.2 quasinorm equivalence of several maximal functions

6.4.3 consequences of the characterizations of hardy spaces

6.4.4 vector-valued hp and its characterizations

6.4.5 singular integrals on hardy spaces

6.4.6 the littlewood-paley characterization of hardy spaces

exercises

 6.5 besov-lipschitz and triebel-lizorkin spaces

6.5.1 introduction of function spaces

6.5.2 equivalence of definitions

exercises

 6.6 atomic decomposition

6.6.1 the space of sequences fa,qp

6.6.2 the smooth atomic decomposition of fa,q

6.6.3 the nonsmooth atomic decomposition of fa,q

6.6.4 atomic decomposition of hardy spaces

exercises

 6.7 singular integrals on function spaces

6.7.1 singular integrals on the hardy space ht

6.7.2 singular integrals on besov-lipschitz spaces

6.7.3 singular integrals on hp(rn)

6.7.4 a singular integral characterization ofh1 (rn)

exercises

7 bmo and carleson measures

 7.1 functions of bounded mean oscillation

7.1.1 definition and basic properties of bmo

7.1.2 the john-nirenberg theorem

7.1.3 consequences of theorem 7.1.6

exercises

 7.2 duality between h1 and bmo

exercises

 7.3 nontangential maximal functions and carleson measures

7.3.1 definition and basic properties of carleson measures

7.3.2 bmo functions and carleson measures

exercises

 7.4 the sharp maximal function

7.4.1 definition and basic properties of the sharp maximal function

7.4.2 a good lambda estimate for the sharp function

7.4.3 interpolation using bmo

7.4.4 estimates for singular integrals involving the sharp function

exercises

 7.5 commutators of singular integrals with bmo functions

7.5.1 an orlicz-type maximal function

7.5.2 a pointwise estimate for the commutator

7.5.3 lp boundedness of the commutator

exercises z

8 singular integrals of nonconvolution type

 8.1 general background and the role of bmo

8.1.1 standard kernels

8.1.2 operators associated with standard kernels

8.1.3 calder6n-zygmund operators acting on bounded functions

exercises

 8.2 consequences of l2 boundedness

8.2.1 weaktype (1, i) and/_,p boundedness of singular integrals

8.2.2 boundedness of maximal singular integrals

8.2.3 h1 → l1 and l∞→bmo boundedness of singular integrals

exercises

 8.3 the t(1) theorem

8.3.1 preliminaries and statement of the theorem

8.3.2 the proof of theorem 8.3.3

8.3.3 an application

exercises

 8.4 paraproducts

8.4.1 introduction to paraproducts

8.4.2 l2 boundedness of paraproducts

8.4.3 fundamental properties of paraproducts

exercises

 8.5 an almost orthogonality lemma and applications

8.5.1 the cotlar-knapp-stein almost orthogonality lemma

8.5.2 an application

8.5.3 almost orthogonality and the t(1) theorem

8.5.4 pseudodifferential operators

exercises

 8.6 the cauchy integral of caldertn and the t(b) theorem

8.6.1 introduction of the cauchy integral operator along a lipschitz curve

8.6.2 resolution of the cauchy integral and reduction of its l2 boundedness to a quadratic estimate

8.6.3 a quadratic t(1) type theorem

8.6.4 a t(b) theorem and the l2 boundedness of the cauchy integral

exercises

 8.7 square roots of elliptic operators

8.7.1 preliminaries and statement of the main result

8.7.2 estimates for elliptic operators on rn

8.7.3 reduction to a quadratic estimate

8.7.4 reduction to a carleson measure estimate

8.7.5 the t(b) argument

8.7.6 the proof of lemma 8.7.9

exercises

9 weighted inequalities

 9.1 the at, condition

9.1.1 motivation for the at, condition

9.1.2 properties of at, weights

exercises

 9.2 reverse htlder inequality and consequences

9.2.1 the reverse helder property of at, weights

9.2.2 consequences of the reverse holder property

exercises

 9.3 the a∞ condition

9.3.1 the class of a∞ weights

9.3.2 characterizations of a∞ weights

exercises

 9.4 weighted norm inequalities for singular integrals

9.4.1 a review of singular integrals

9.4.2 a good lambda estimate for singular integrals

9.4.3 consequences of the good lambda estimate

9.4.4 necessity of the at, condition

exercises

 9.5 further properties of ap weights

9.5.1 factorization of weights

9.5.2 extrapolation from weighted estimates on a single d~0

9.5.3 weighted inequalities versus vector-valued inequalities

exercises

10 boundedness and convergence of fourier integrals

 10.1 the multiplier problem for the ball

10.1.1 sprouting of triangles

10.1.2 the counterexample

exercises

 10.2 bochner-riesz means and the carleson-sjolin theorem

10.2.1 the bochner-riesz kernel and simple estimates

10.2.2 the carleson-sj01in theorem

10.2.3 the kakeya maximal function

10.2.4 boundedness of a square function

10.2.5 the proof of lemma 10.2.5

exercises

 10.3 kakeya maximal operators

10.3.1 maximal functions associated with a set of directions

10.3.2 the boundedness of σn on lp(r2)

10.3.3 the higher-dimensional kakeya maximal operator

exercises

 10.4 fourier transform restriction and bochner-riesz means

10.4.1 necessary conditions for rp→q(sn-1) to hold

10.4.2 a restriction theorem for the fourier transform

10.4.3 applications to bochner-riesz multipliers

10.4a the full restriction theorem on r2

exercises

 10.5 almost everywhere convergence of bochner-riesz means

10.5.1 a counterexample for the maximal bochner-riesz operator

10.5.2 almost everywhere summability of the bochner-riesz means

10.5.3 estimates for radial multipliers

exercises

11 time--frequency analysis and the carleson-hunt theorem

 11.1 almost everywhere convergence of fourier integrals

11.1.1 preliminaries

11.1.2 discretization of the carleson operator

11.1.3 linearization of a maximal dyadic sum

11.1.4 iterative selection of sets of tiles with large mass and

energy

11.1.5 proof of the mass lemma 11.1.8

11.1.6 proof of energy lemma 11.1.9

11.1.7 proof of the basic estimate lemma 11.1.10

exercises

 11.2 distributional estimates for the carleson operator

1.2.1 the main theorem and preliminary reductions

11.2.2 the proof of estimate (11.2.8)

11.2.3 the proof of estimate (11.2.9)

11.2.4 the proof of lemma 11.2.2

exercises

 11.3 the maximal carleson operator and weighted estimates

exercises

glossary

references

index