索伯列夫乘子理论

introduction

part i description and properties of multipliers

1 trace inequalities for functions in sobolev spaces.

1.1 trace inequalities for functions in wm1 and wm1

1.2 trace inequalities for functions in wmp and wmp， p＞1

1.3 estimate for the lq-norm with respect to an arbitrary measure

2 multipliers in pairs of sobolev spaces

2.1 introduction

2.2 characterization of the space m（wm1 → wl1）

2.3 characterization of the space m（wmp → wlp） for p＞1

2.4 the space m（wmp（rn+）→wlp（rn+））

2.5 the space m（wmp→w-kp）

2.6 the space m（wmp→wlp）

2.7 certain properties of multipliers

2.8 the space m（wmp→wlp）

2.9 multipliers in spaces of functions with bounded variation.

3 multipliers in pairs of potential spaces

3.1 trace inequality for bessel and riesz potential spaces

3.2 description of m（hmp→hlp）

.3.3 one-sided estimates for the norm in m（hmp→hlp）

3.4 upper estimates for the norm in m（hmp→hlp）by norms in besov spaces

3.5 miseenaneous properties of multipliers in m（hmp→hlp）

3.6 spectrum of multipliers in hlp and h-lp'

3.7 the space m（hmp→hlp）

3.8 positive homogeneous multipliers

4 the space m（bmp→blp） with p＞1

4.1 introduction

4.2 properties of besov spaces

4.3 proof of theorem 4.1.1

4.4 sufficient conditious for inclusion into m（wmp→wlp）with noninteger m and l

4.5 conditions involving the space hln/m.

4.6 composition operator on m（wmp→wlp）

5 the space m（bm1→bl1）

5.1 trace inequality for functions in bl1（rn）

5.2 properties of functions in the space bk1（rn） ，

5.3 descriptions of-m（bm1→bl1） with integer l

5.4 description of the space-m（bm1→bl1） with noninteger l

5.5 further results on multipliers in besov and other function spaces

6 maximal algebras in spaces of multipliers

6.1 introduction

6.2 pointwise interpolation inequalities for derivatives

6.3 maximal banach algebra in m（wmp→wlp）

6.4 maximal algebra in spaces of bessel potentials

6.5 imbeddings of maximal algebras

7 essential norm and compactness of multipliers

7.1 auxiliary assertions

7.2 two-sided estimates for the essential norm. the case m＞l

7.3 two-sided estimates for the essential norm in the case m = l

8.traces and extensions of multipliers

8.1 introduction

8.2 multipliers in pairs of weighted sobolev spaces in rn+

8.3 characterization of m（wpt，→wps，）

8.4 auxiliary estimates for an extension operator

8.5 trace theorem fo/the space m（wpt，→wps，

8.6 traces of multipliers on the smooth boundary of a domain.

8.7 mwlp（rn） as the space of traces of multipliers in the weighted sobolev space wp，k（r+n+1）

8.8 traces of functions in mwpl（rn+m） on rn

8.9 multipliers in the space of bessel potentials as traces of multipliers

9 sobolev multipliers in a domain， multiplier mappings and manifolds

9.1 multipliers in a special lipschitz domain

9.2 extension of multipliers to the complement of a special lipschitz domain

9.3 multipliers in a bounded domain

9.4 change of variables in norms of sobolev spaces

9.5 implicit function theorems

9.6 space

part ii applications of multipliers to differential and integral operators

10 differential operators in pairs of sobolev spaces

10.1 the norm of a differential operator： wph→wph-k

10.2 essential norm of a differential operator

10.3 fredholm property of the schr6dinger operator

10.4 domination of differential operators in rn

11 schrsdinger operator and m（w21→w2-1）

11.1 introduction

11.2 characterization of m（w21→w2-1） and the schrodinger operator on w12

11.3 a compactness criterion

11.4 characterization of m（w21→w2-1）

11.5 characterization of the space m（w21（）→w2-1（））

11.6 second-order differential operators acting from w21 to w21

12 relativistic schrsdinger operator and m（w21/2→w21/2）

12.1 auxiliary assertions

12.2 corollaries of the form boundedness criterion and related results

13 multipliers as solutions to elliptic equations

13.1 the dirichlet problem for the linear second-order-elliptic equation in the space of multipliers

13.2 bounded solutions of linear eniptic equations as multipliers

13.3 solvability of quasilinear elliptic equations in spaces of multipliers

13.4 coercive estimates for solutions of elliptic equations in spaces of multipliers

13.5 smoothness of solutions to higher order elliptic semilinear systems

14 regularity of the boundary in lv-theory of elliptic boundary value problems

14.1 description of results

14.2 change of variables in differential operators

14.3 fredholm property of the elliptic b?undary value problem

14.4 auxiliary assertions

14.5 solvability of the dirichlet problem in wlp（）

14.6 necessity of assumptions on the domain

14.7 local characterization of mpl-1/p（）

15 multipliers in the classical layer potential theory for lipschitz domains

15.1 introduction

15.2 solvability of boundary value problems in weighted sobolev spaces

15.3 continuity properties of boundary integral operators

15.4 proof of theorems 15.1.1 and 15.1.2

15.5 properties of surfaces in the class mpl（）

15.6 sharpness of conditions imposed on

15.7 extension to boundary integral equations of elasticity

16 applications of multipliers to the theory of integral operators

16.1 convolution operator in weighted l2-spaces

16.2 calculus of singular integral operators with symbols in spaces of multipliers

16.3 continuity in sobolev spaces of singular integral operators with symbols depending on x

references

list of symbols

author and subject index

《索伯列夫乘子理论》旨在为读者全面讲述微分函数空间对中点乘子理论。这个理论是在过去的三十年中通过众多学者大量积累发展起来的，《索伯列夫乘子理论》是前人结果的延伸和扩展。第一部分介绍了乘子理论，囊括了众多理论和概念，如，迹不等式、乘子的解析特性、索伯列夫乘子空间和其他空间之间的关系、乘子空间最大子代数、迹和乘子扩展、乘子的范数和紧性以及乘子的综合特性；第二部分包括了该理论的大量应用，索伯列夫空间对中微分算子的连续性和紧性；乘子作为线性和伪线性双曲方程的解；lipschitz域中单层和双层势能理论的高级正则性和双曲边界值问题l_p理论中边界正则性；索伯列夫空间中的奇异积分算子。这部著作综合性强，文笔流畅，结构紧凑，是泛函分析，偏微分方程和伪微分算子等相关数学专业不可多得的教材和参考书。