索伯列夫乘子理论

　　《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。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*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。

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