索伯列夫乘子理論

前言

　　'I never heard of "Uglifieation，"Alice ventured to say. 'What is. it?'' 　　Lewis Carroll，"Alice in Wonderland" 　　Subject and motivation. The present book is devoted to a theory of mul-tipliers in spaces of differentiable functions and its applications to analysis，partial differential and integral equations. By a multiplier acting from one function space S1 into another S2， we mean a function which defines a bounded linear mapping of $1 into S2 by pointwise multiplication. Thus with any pair of spaces S1， S2 we associate a third one， the space of multipliers M（S1→S2） endowed with the norm of the operator of multiplication. In what follows， the role of the spaces S1 and $2 iq. played by Sobolev spaces， Bessel potential spaces， Besov spaces， and the like. 　　The Fourier multipliers are not dealt with in this book. In order to empha-size the difference between them and the multipliers under consideration， we attach Sobolev's name to the latter. By coining the term Sobolev multipliers we just hint at various spaces of differentiable functions of Sobolev's type，being fully aware that Sobolev never worked on multipliers. After all， Fourier never did either. 　　Sobolev multipliers arise in many problems of analysis and theories of par-tial differential and integral equations. Coefficients of differential operators can be naturally considered as multipliers. The same is true for symbols of more general pseudo-differential operators. Multipliers also appear in the theory of differentiable mappings preserving Sobolev spaces. Solutions of boundary value problems can be sought in classes of multipliers. Because of their aL gebraic properties， multipliers are suitable objects for generalizations of the basic facts of calculus （theorems on implicit functions， traces and extensions，point mappings and their compositions etc.） Moreover， some basic operators of harmonic analysis， like the classical maximal and singular integral opera-tors， act in certain classes of multipliers. 　　We believe that the calculus of Sobolev multipliers provides an adequate language for future work in the theory of linear and nonlinear differential and pseudodifferential equations under minimal restrictions on the coefficients， 　　domains， and other data. 　　Before the 1970s， the word multiplier was usually associated with the name of Fourier， and a deep theory of Lp-Fourier multipliers created by Marcinkiewicz， Mikhlin， HSrmander et al was quite popular. As for the multi- pliers preserving a space of differentiable functions， only a few isolated results were known （Devinatz and Hirschman [DH]， Hirschman [Hi1]， [Hi2]， Strichartz [Str]， Polking [Poll]， Peetre [Pe2]）， while the multipliers in pairs of such spaces were not considered at all. 　　The first （and the only one for the time being） attempt to work out a more or less comprehensive theory of multipliers acting either in one or in a pair of spaces of Sobolev type was undertaken by the authors in the late 1970s and early 1980s [Maz10]， [Maz12]， [MSh1]-[MSh16]. Results of that theory were collected in our monograph "Theory of Multipliers in Spaces of Differ- entiable Functions" （Pitman， 1985） [MSh16]. During the last two decades， we continued to work in the area， adding new results and developing further ap-plications [Sh2]-[Sh14]， [MSh17]-[MSh23]. We wish to reflect the present state of our theory in this book. An essential part of the aforementioned monograph is also included here. 　　No results concerning multipliers in spaces of analytic functions are men-tioned in what follows， in contrast to [MShl6]. To describe progress in this area achieved during the last twenty five years would require a disproportion-ate growth of the book. 　　Structure of the book. The book consists of two parts. Part I is devoted to the theory of multipliers and covers the following topics： 　　Trace inequalities 　　Analytic characterization of multipliers 　　Relations between spaces of Sobolev multipliers and other function spaces 　　Maximal subalgebras of multiplier spaces 　　Traces and extensions of multipliers 　　Essential norm and compactness of multipliers 　　Miscellaneous properties of multipliers （spectrum， composition and im-plicit function theorems， point mappings preserving Sobolev spaces， etc.） 　　In Part II we dwell upon several applications of this theory. Their list is as follows： 　　Continuity and compactness of differential operators in pairs of Sobolev spaces 　　Multipliers as solutions to linear and quasilinear elliptic equations Higher regularity in the single and double layer potential theory for Lipschitz domains 　　B，egularity of the boundary in Lp-theory of elliptic boundary value prob- lems 　　Singular integral operators in Sobolev spaces 　　Each chapter starts with a short introductory outline of the included material. 　　Readership. The volume is addressed to mathematicians working in func-tional analysis and in the theories of partial differential， integral， and pseudo-differential operators. Prerequisites for reading this book are undergraduate courses in these subjects. 　　Acknowledgments. V. Maz'ya was partially supported by the Na-tional Science Foundation （Grant DMS-0500029， USA） and EPSRC （Grant EP/F005563/1， UK）. T. Shaposhnikova gratefully acknowledges support from the Swedish Natural Science Research Council （VR）.

內(nèi)容概要

　　《索伯列夫乘子理論》旨在為讀者全面講述微分函數(shù)空間對中點乘子理論。這個理論是在過去的三十年中通過眾多學者大量積累發(fā)展起來的，《索伯列夫乘子理論》是前人結(jié)果的延伸和擴展。第一部分介紹了乘子理論，囊括了眾多理論和概念，如，跡不等式、乘子的解析特性、索伯列夫乘子空間和其他空間之間的關(guān)系、乘子空間最大子代數(shù)、跡和乘子擴展、乘子的范數(shù)和緊性以及乘子的綜合特性；第二部分包括了該理論的大量應用，索伯列夫空間對中微分算子的連續(xù)性和緊性；乘子作為線性和偽線性雙曲方程的解；lipschitz域中單層和雙層勢能理論的高級正則性和雙曲邊界值問題l_p理論中邊界正則性；索伯列夫空間中的奇異積分算子。這部著作綜合性強，文筆流暢，結(jié)構(gòu)緊湊，是泛函分析，偏微分方程和偽微分算子等相關(guān)數(shù)學專業(yè)不可多得的教材和參考書。

作者簡介

劉文勇，中國人民大學博士，嚴苛的托福閱讀、SAT寫作教師，原新東方集團培訓師，現(xiàn)任樂聞攜爾教育咨詢有限公司總裁兼創(chuàng)始人。他是托?？忌貍鋮⒖假Y料——《新托福黃金精選閱讀》的整理者，也是《新托福真題詳解系列》以及《去美國讀本科》的書籍主編。 常以老派的知識分子自居，高度近視，精力充沛，熱愛紅牛。自詡為渾身贅肉而有問必答的熱心人士。大學時代他就開始了兼職英語老師的職業(yè)生涯，講授包括托福在內(nèi)的諸多出國留學考試輔導課程。 他講課激情四射、風趣幽默，在強調(diào)英語綜合能力的同時，擅長引導學生找到屬于自己的解題技巧與思考方式，并將這些方法升華且融化到實際的做題之中去，已成功幫助上萬名學生實現(xiàn)了解題思路“從無到有，從有到無”的質(zhì)的飛躍。

書籍目錄

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

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