Theory of Sobolev multipliers: with applications to differential and integral operators
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| Sprache: | English |
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Berlin [u.a.]
Springer
2009
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| Schriftenreihe: | Grundlehren der mathematischen Wissenschaften
337 |
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| Beschreibung: | XIII, 609 S. 235 mm x 155 mm |
| ISBN: | 9783540694908 |
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| 100 | 1 | |a Mazʹja, Vladimir Gilelevič |d 1937- |e Verfasser |0 (DE-588)121490602 |4 aut | |
| 245 | 1 | 0 | |a Theory of Sobolev multipliers |b with applications to differential and integral operators |c Vladimir G. Maz'ya ; Tatyana O. Shaposhnikova |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2009 | |
| 300 | |a XIII, 609 S. |c 235 mm x 155 mm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Grundlehren der mathematischen Wissenschaften |v 337 | |
| 650 | 0 | 7 | |a Multiplikator |0 (DE-588)4040703-2 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Sobolev-Raum |0 (DE-588)4055345-0 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Sobolev-Raum |0 (DE-588)4055345-0 |D s |
| 689 | 0 | 1 | |a Multiplikator |0 (DE-588)4040703-2 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Šapošnikova, T. O. |d 1946- |e Verfasser |0 (DE-588)13670493X |4 aut | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-3-540-69492-2 |
| 830 | 0 | |a Grundlehren der mathematischen Wissenschaften |v 337 |w (DE-604)BV000000395 |9 337 | |
| 856 | 4 | 2 | |m HBZ Datenaustausch |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016771204&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
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Datensatz im Suchindex
| _version_ | 1804138075016658944 |
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| adam_text | Contents
Introduction................................................... 1
Part I Description and Properties of Multipliers
1 Trace Inequalities for Functions in Sobolev Spaces......... 7
1.1 Trace Inequalities for Functions in w 1 and W[n............. 7
1.1.1 The Case m = 1................................... 7
1.1.2 The Case m 1................................... 12
1.2 Trace Inequalities for Functions in w™ and W™. p 1....... 14
1.2.1 Preliminaries ..................................... 14
1.2.2 The (p, m)-Capacity............................... 16
1.2.3 Estimate for the Integral of Capacity of a Set Bounded
by a Level Surface................................. 19
1.2.4 Estimates for Constants in Trace Inequalities......... 22
1.2.5 Other Criteria for the Trace Inequality (1.2.29)
with p 1........................................ 25
1.2.6 The Fefferman and Phong Sufficient Condition........ 28
1.3 Estimate for the Lg-Norm with respect to an Arbitrary
Measure................................................ 29
1.3.1 The case 1 p q................................ 30
1.3.2 The case q p n/m ............................. 30
2 Multipliers in Pairs of Sobolev Spaces..................... 33
2.1 Introduction............................................ 33
2.2 Characterization of the Space M(W{ -» W[)............... 35
2.3 Characterization of the Space M(W™ - Wlp) for p 1....... 38
2.3.1 Another Characterization of the Space A/(Wpm - Wlp)
for 0 / m, pm n, p 1 ....................... 43
2.3.2 Characterization of the Space M{W™ -* Wlp)
for pm n, p 1 ................................. 47
VI Contents
2.3.3 One-Sided Estimates for Norms of Multipliers
in the Case pm n................................ 48
2.3.4 Examples of Multipliers............................ 49
2.4 The Space M(W™(R£) -? Wj(R£)) ....................... 50
2.4.1 Extension from a Half-Space........................ 50
2.4.2 The Case p 1 ................................... 51
2.4.3 The Case p=l ................................... 53
2.5 The Space M{W™ -? Wpk) .............................. 54
2.6 The Space M(W™ - Wlq)................................ 57
2.7 Certain Properties of Multipliers.......................... 58
2.8 The Space M(wj™ - wlp)................................. 60
2.9 Multipliers in Spaces of Functions with Bounded Variation. ... 63
2.9.1 The Spaces Mbv and MBV ........................ 66
3 Multipliers in Pairs of Potential Spaces.................... 69
3.1 Trace Inequality for Bessel and Riesz Potential Spaces ....... 69
3.1.1 Properties of Bessel Potential Spaces................. 70
3.1.2 Properties of the (p, m)-Capacity.................... 71
3.1.3 Main Result...................................... 73
3.2 Description of M{H™ - Hlp) ............................. 75
3.2.1 Auxiliary Assertions............................... 75
3.2.2 Imbedding of M(H? -+ Hlp) into M(H?~l - Lp)..... 76
3.2.3 Estimates for Derivatives of a Multiplier.............. 78
3.2.4 Multiplicative Inequality for the Strichartz Function ... 79
3.2.5 Auxiliary Properties of the Bessel Kernel Gi.......... 80
3.2.6 Upper Bound for the Norm of a Multiplier............ 81
3.2.7 Lower Bound for the Norm of a Multiplier............ 85
3.2.8 Description of the Space M(H™ -? Hlp).............. 86
3.2.9 Equivalent Norm in M(H™ - Hlp) Involving
the Norm in I/mp/(m_/)............................. 87
3.2.10 Characterization of M(H™ - Hlp), m I, Involving
the Norm in Li)Unif................................ 89
3.2.11 The Space M{H™ -»¦ Hlp) for mp n................ 95
3.3 One-Sided Estimates for the Norm in M(#™ - Hlp)......... 95
3.3.1 Lower Estimate for the Norm in M(#™ - Hlp)
Involving Morrey Type Norms...................... 96
3.3.2 Upper Estimate for the Norm in M(H™ - Hlp)
Involving Marcinkiewicz Type Norms................ 96
3.3.3 Upper Estimates for the Norm in M(#™ - Hl)
Involving Norms in Hln/m .......................... 98
3.4 Upper Estimates for the Norm in Mill™ - Hlp) by Norms
in Besov Spaces......................................... 99
3.4.1 Auxiliary Assertions............................... 99
3.4.2 Properties of the Space B% oo ....................... 103
Contents VII
3.4.3 Estimates for the Norm in M{Hpn - Hlp)
by the Norm in B^ ..............................108
3.4.4 Estimate for the Norm of a Multiplier in Jl/i/^R1)
by the (/-Variation.................................110
3.5 Miscellaneous Properties of Multipliers in M{H™ - H p).....Ill
3.6 Spectrum of Multipliers in Hlp and Hp,1 ....................115
3.6.1 Preliminary Information............................115
3.6.2 Facts from Nonlinear Potential Theory...............117
3.6.3 Main Theorem....................................118
3.6.4 Proof of Theorem 3.6.1.............................120
3.7 The Space M(/i™ - h!p)..................................122
3.8 Positive Homogeneous Multipliers .........................125
3.8.1 The Space M{Hpn(dBx) - Hp{dB^)) ................125
3.8.2 Other Normalizations of the Spaces h pn and Hp .......127
3.8.3 Positive Homogeneous Elements of the Spaces
M(h » - h p) and M(H? - Hlp)....................130
The Space M(B™ - Blp) with p 1.......................133
4.1 Introduction............................................133
4.2 Properties of Besov Spaces................................134
4.2.1 Survey of Known Results...........................134
4.2.2 Properties of the Operators Dp,/ and Dpj ............136
4.2.3 Pointwise Estimate for Bessel Potentials..............138
4.3 Proof of Theorem 4.1.1...................................141
4.3.1 Estimate for the Product of First Differences..........141
4.3.2 Trace Inequality for B*, p 1 ......................143
4.3.3 Auxiliary Assertions Concerning A/(B™ — Blp) .......145
4.3.4 Lower Estimates for the Norm in M{B1pn -+ Blp).......146
4.3.5 Proof of Necessity in Theorem 4.1.1..................149
4.3.6 Proof of Sufficiency in Theorem 4.1.1 ................155
4.3.7 The Case rap n .................................164
4.3.8 Lower and Upper Estimates for the Norm
in M(B™ - B p)..................................165
4.4 Sufficient Conditions for Inclusion into M(W™ — Wlp)
with Noninteger m and / .................................166
4.4.1 Conditions Involving the Space B£ x ................166
4.4.2 Conditions Involving the Fourier Transform...........168
4.4.3 Conditions Involving the Space Blqp .................170
4.5 Conditions Involving the Space Hln, ......................173
4.6 Composition Operator on M{W™ - Wl)...................174
n/ri
•p ~^ pi
VIII Contents
5 The Space M(B™ — B[)..................................179
5.1 Trace Inequality for Functions in B (Rn)...................179
5.1.1 Auxiliary Facts ...................................180
5.1.2 Main Result......................................183
5.2 Properties of Functions in the Space B (Kn)................185
5.2.1 Trace and Imbedding Properties.....................185
5.2.2 Auxiliary Estimates for the Poisson Operator.........189
5.3 Descriptions of M(Bf -+ B ) with Integer I ................193
5.3.1 A Norm in M{B? - B[) ..........................194
5.3.2 Description of M(B? - B[) Involving Du...........199
5.3.3 M{B^{Wl) - B[{Rn)) as the Space of Traces........201
5.3.4 Interpolation Inequality for Multipliers...............202
5.4 Description of the Space M{Bf - B[) with Noninteger I ___203
5.5 Further Results on Multipliers in Besov and Other Function
Spaces ................................................. 206
5.5.1 Peetre s Imbedding Theorem........................206
5.5.2 Related Results on Multipliers in Besov
and Triebel-Lizorkin Spaces.........................208
5.5.3 Multipliers in BMO...............................210
6 Maximal Algebras in Spaces of Multipliers ................213
6.1 Introduction............................................213
6.2 Pointwise Interpolation Inequalities for Derivatives...........214
6.2.1 Inequalities Involving Derivatives of Integer Order.....214
6.2.2 Inequalities Involving Derivatives of Fractional Order ..215
6.3 Maximal Banach Algebra in M(W™ - Wlp) ................220
6.3.1 The Case p 1 ...................................220
6.3.2 Maximal Banach Algebra in M(Wp -»¦ W[) ..........224
6.4 Maximal Algebra in Spaces of Bessel Potentials .............227
6.4.1 Pointwise Inequalities Involving the Strichartz
Function.........................................227
6.4.2 Banach Algebra A™ 1..............................231
6.5 Imbeddings of Maximal Algebras..........................233
7 Essential Norm and Compactness of Multipliers...........241
7.1 Auxiliary Assertions.....................................243
7.2 Two-Sided Estimates for the Essential Norm. The Case
m I..................................................248
7.2.1 Estimates Involving Cutoff Functions................248
7.2.2 Estimate Involving Capacity (The Case
mp n, p 1)....................................250
7.2.3 Estimates Involving Capacity (The Case
mp = n, p 1)....................................257
7.2.4 Proof of Theorem 7.0.3 ............................261
7.2.5 Sharpening of the Lower Bound for the Essential
Norm in the Case m /, mp n, p 1..............262
Contents IX
7.2.6 Estimates of the Essential Norm for mp n, p 1
and for p = 1 .....................................263
7.2.7 One-Sided Estimates for the Essential Norm..........266
7.2.8 The Space of Compact Multipliers...................267
7.3 Two-Sided Estimates for the Essential Norm
in the Case m = I .......................................270
7.3.1 Estimate for the Maximum Modulus of a Multiplier
in Wp by its Essential Norm........................270
7.3.2 Estimates for the Essential Norm Involving Cutoff
Functions (The Case Ip n, p 1)..................272
7.3.3 Estimates for the Essential Norm Involving Capacity
(The Case Ip n, p 1)...........................277
7.3.4 Two-Sided Estimates for the Essential Norm
in the Cases Ip n, p 1, and p = ................278
7.3.5 Essential Norm in MWlp............................281
Traces and Extensions of Multipliers.......................285
8.1 Introduction............................................285
8.2 Multipliers in Pairs of Weighted Sobolev Spaces in W _ .......285
8.3 Characterization of M{W£(i - W°-a)......................288
8.4 Auxiliary Estimates for an Extension Operator..............292
8.4.1 Pointwise Estimates for T-y and VT-y................292
8.4.2 Weighted Lp-Estimates for T-y and VT7 .............294
8.5 Trace Theorem for the Space M(W^ -» W°-a) .............297
8.5.1 The Case l ....................................298
8.5.2 The Case l ....................................301
8.5.3 Proof of Theorem 8.5.1 for I 1.....................303
8.6 Traces of Multipliers on the Smooth Boundary of a Domain. . . 304
8.7 MWj,(Rn) as the Space of Traces of Multipliers in the
Weighted Sobolev Space W£0{Rn+m)......................305
8.7.1 Preliminaries .....................................305
8.7.2 A Property of Extension Operator..................306
8.7.3 Trace and Extension Theorem for Multipliers.........308
8.7.4 Extension of Multipliers from Rn to R +1 ............311
8.7.5 Application to the First Boundary Value Problem
in a Half-Space....................................311
8.8 Traces of Functions in i /W^(Kn+m) on K .................312
8.8.1 Auxiliary Assertions...............................313
8.8.2 Trace and Extension Theorem ......................315
8.9 Multipliers in the Space of Bessel Potentials as Traces
of Multipliers...........................................319
8.9.1 Bessel Potentials as Traces .........................319
8.9.2 An Auxiliary Estimate for the Extension Operator T .. 320
8.9.3 MHlp as a Space of Traces..........................322
Contents
Sobolev Multipliers in a Domain, Multiplier Mappings
and Manifolds.............................................325
9.1 Multipliers in a Special Lipschitz Domain...................326
9.1.1 Special Lipschitz Domains..........................326
9.1.2 Auxiliary Assertions...............................326
9.1.3 Description of the Space of Multipliers...............329
9.2 Extension of Multipliers to the Complement of a Special
Lipschitz Domain........................................332
9.3 Multipliers in a Bounded Domain..........................336
9.3.1 Domains with Boundary in the Class C01............336
9.3.2 Auxiliary Assertions...............................337
9.3.3 Description of Spaces of Multipliers in a Bounded
Domain with Boundary in the Class C0 1.............339
9.3.4 Essential Norm and Compact Multipliers
in a Bounded Lipschitz Domain.....................340
9.3.5 The Space MLp(tt) for an Arbitrary
Bounded Domain..................................346
9.4 Change of Variables in Norms of Sobolev Spaces.............350
9.4.1 (p, /)-Diffeomorphisms..............................350
9.4.2 More on (p, Z)-Diffeomorphisms......................352
9.4.3 A Particular (p, Z)-Diffeomorphism...................353
9.4.4 (p, O-Manifolds....................................356
9.4.5 Mappings T™1 of One Sobolev Space into Another___357
9.5 Implicit Function Theorems...............................364
9.6 The Space M{W™{Q) - Wlp{Q)) .........................367
9.6.1 Auxiliary Results..................................367
9.6.2 Description of the Space M{W™{fl) - Wp{H)).......369
Part II Applications of Multipliers to Differential and Integral
Operators
10 Differential Operators in Pairs of Sobolev Spaces..........373
10.1 The Norm of a Differential Operator: Wfi -* W^~k..........373
10.1.1 Coefficients of Operators Mapping Wp into Wph~k
as Multipliers.....................................374
10.1.2 A Counterexample.................................378
10.1.3 Operators with Coefficients Independent
of Some Variables.................................379
10.1.4 Differential Operators on a Domain..................382
10.2 Essential Norm of a Differential Operator...................384
10.3 Fredholm Property of the Schrodinger Operator.............386
10.4 Domination of Differential Operators in E™.................387
Contents XI
11 Schrodinger Operator and M(w* —? w^1) .................391
11.1 Introduction............................................391
11.2 Characterization of M{w — w^1) and the Schrodinger
Operator on wi, .........................................393
11.3 A Compactness Criterion.................................407
11.4 Characterization of M(Wj - W2l) .......................411
11.5 Characterization of the Space M(w (Q) —* w^1^)).........416
11.6 Second-Order Differential Operators Acting from w to w^1 . . 421
12 Relativistic Schrodinger Operator
and M(W21/2 - W^1 2)...................................427
12.1 Auxiliary Assertions.....................................427
12.1.1 Main Result......................................436
12.2 Corollaries of the Form Boundedness Criterion and Related
Results.................................................441
13 Multipliers as Solutions to Elliptic Equations..............445
13.1 The Dirichlet Problem for the Linear Second-Order Elliptic
Equation in the Space of Multipliers.......................445
13.2 Bounded Solutions of Linear Elliptic Equations
as Multipliers...........................................447
13.2.1 Introduction......................................447
13.2.2 The Case /? 1...................................448
13.2.3 The Case /3 = 1...................................452
13.2.4 Solutions as Multipliers from W^w{p){Q) into W^O) .454
13.3 Solvability of Quasilinear Elliptic Equations in Spaces
of Multipliers...........................................456
13.3.1 Scalar Equations in Divergence Form ................457
13.3.2 Systems in Divergence Form........................458
13.3.3 Dirichlet Problem for Quasilinear Equations
in Divergence Form................................461
13.3.4 Dirichlet Problem for Quasilinear Equations
in Nondivergence Form.............................463
13.4 Coercive Estimates for Solutions of Elliptic equations
in Spaces of Multipliers ..................................467
13.4.1 The Case of Operators in Rn .......................467
13.4.2 Boundary Value Problem in a Half-Space.............469
13.4.3 On the Loo-Norm in the Coercive Estimate...........473
13.5 Smoothness of Solutions to Higher Order Elliptic Semilinear
Systems................................................474
13.5.1 Composition Operator in Classes of Multipliers .......474
13.5.2 Improvement of Smoothness of Solutions to Elliptic
Semilinear Systems................................477
XII Contents
14 Regularity of the Boundary in Lp-Theory of Elliptic
Boundary Value Problems.................................479
14.1 Description of Results....................................479
14.2 Change of Variables in Differential Operators...............481
14.3 Fredholm Property of the Elliptic Boundary Value Problem . . . 483
14.3.1 Boundaries in the Classes Mp~1/p, Wp~1/p,
and Mlp l/p{5)....................................483
14.3.2 A Priori Lp-Estimate for Solutions and Other
Properties of the Elliptic Boundary Value Problem .... 484
14.4 Auxiliary Assertions.....................................489
14.4.1 Some Properties of the Operator T..................489
14.4.2 Properties of the Mappings A and x.................490
14.4.3 Invariance of the Space Wlp n Wp Under a Change
of Variables.......................................492
14.4.4 The Space W~k for a Special Lipschitz Domain.......496
14.4.5 Auxiliary Assertions on Differential Operators
in Divergence Form................................498
14.5 Solvability of the Dirichlet Problem in Wlp(Q)...............502
14.5.1 Generalized Formulation of the Dirichlet Problem.....502
14.5.2 A Priori Estimate for Solutions of the Generalized
Dirichlet Problem.................................502
14.5.3 Solvability of the Generalized Dirichlet Problem.......503
14.5.4 The Dirichlet Problem Formulated in Terms of Traces. . 504
14.6 Necessity of Assumptions on the Domain...................507
14.6.1 A Domain Whose Boundary is in M%/2 n C1
but does not Belong to M%/2(6).....................507
14.6.2 Necessary Conditions for Solvability of the Dirichlet
Problem..........................................509
14.6.3 Boundaries of the Class Mp~1/p(S)...................510
14.7 Local Characterization of Mp~1/p( 5).......................513
14.7.1 Estimates for a Cutoff Function.....................513
14.7.2 Description of Mp~1/p(6) Involving a Cutoff Function . . 515
14.7.3 Estimate for si....................................516
14.7.4 Estimate for s2....................................520
14.7.5 Estimate for s3....................................523
15 Multipliers in the Classical Layer Potential Theory
for Lipschitz Domains .....................................531
15.1 Introduction............................................531
15.2 Solvability of Boundary Value Problems in Weighted Sobolev
Spaces.................................................537
15.2.1 (p, k, a)-Diffeomorphisms...........................537
15.2.2 Weak Solvability of the Dirichlet Problem............539
15.2.3 Main Result......................................542
Contents XIII
15.3 Continuity Properties of Boundary Integral Operators........547
15.4 Proof of Theorems 15.1.1 and 15.1.2.......................551
15.4.1 Proof of Theorem 15.1.1............................551
15.4.2 Proof of Theorem 15.1.2............................557
15.5 Properties of Surfaces in the Class Mep{5)...................559
15.6 Sharpness of Conditions Imposed on 0(2....................562
15.6.1 Necessity of the Inclusion 0(2 € W^
in Theorem 15.2.1.................................562
15.6.2 Sharpness of the Condition 0(2 E B^ p...............563
15.6.3 Sharpness of the Condition 0(2 G Mp(5)
in Theorem 15.2.1.................................564
15.6.4 Sharpness of the Condition 0(2 £ Mp{S)
in Theorem 15.1.1.................................566
15.7 Extension to Boundary Integral Equations of Elasticity.......568
16 Applications of Multipliers to the Theory of Integral
Operators .................................................573
16.1 Convolution Operator in Weighted Z/2-Spaces...............573
16.2 Calculus of Singular Integral Operators with Symbols
in Spaces of Multipliers ..................................575
16.3 Continuity in Sobolev Spaces of Singular Integral Operators
with Symbols Depending oni ............................579
16.3.1 Function Spaces..................................580
16.3.2 Description of the Space M(Hm- - H *) ...........582
16.3.3 Main Result .....................................585
16.3.4 Corollaries........................................588
References.....................................................591
List of Symbols................................................605
Author and Subject Index.....................................607
|
| adam_txt |
Contents
Introduction. 1
Part I Description and Properties of Multipliers
1 Trace Inequalities for Functions in Sobolev Spaces. 7
1.1 Trace Inequalities for Functions in w"1 and W[n. 7
1.1.1 The Case m = 1. 7
1.1.2 The Case m 1. 12
1.2 Trace Inequalities for Functions in w™ and W™. p 1. 14
1.2.1 Preliminaries . 14
1.2.2 The (p, m)-Capacity. 16
1.2.3 Estimate for the Integral of Capacity of a Set Bounded
by a Level Surface. 19
1.2.4 Estimates for Constants in Trace Inequalities. 22
1.2.5 Other Criteria for the Trace Inequality (1.2.29)
with p 1. 25
1.2.6 The Fefferman and Phong Sufficient Condition. 28
1.3 Estimate for the Lg-Norm with respect to an Arbitrary
Measure. 29
1.3.1 The case 1 p q. 30
1.3.2 The case q p n/m . 30
2 Multipliers in Pairs of Sobolev Spaces. 33
2.1 Introduction. 33
2.2 Characterization of the Space M(W{" -» W[). 35
2.3 Characterization of the Space M(W™ - Wlp) for p 1. 38
2.3.1 Another Characterization of the Space A/(Wpm - Wlp)
for 0 / m, pm n, p 1 . 43
2.3.2 Characterization of the Space M{W™ -* Wlp)
for pm n, p 1 . 47
VI Contents
2.3.3 One-Sided Estimates for Norms of Multipliers
in the Case pm n. 48
2.3.4 Examples of Multipliers. 49
2.4 The Space M(W™(R£) -? Wj(R£)) . 50
2.4.1 Extension from a Half-Space. 50
2.4.2 The Case p 1 . 51
2.4.3 The Case p=l . 53
2.5 The Space M{W™ -? Wpk) . 54
2.6 The Space M(W™ - Wlq). 57
2.7 Certain Properties of Multipliers. 58
2.8 The Space M(wj™ - wlp). 60
2.9 Multipliers in Spaces of Functions with Bounded Variation. . 63
2.9.1 The Spaces Mbv and MBV . 66
3 Multipliers in Pairs of Potential Spaces. 69
3.1 Trace Inequality for Bessel and Riesz Potential Spaces . 69
3.1.1 Properties of Bessel Potential Spaces. 70
3.1.2 Properties of the (p, m)-Capacity. 71
3.1.3 Main Result. 73
3.2 Description of M{H™ - Hlp) . 75
3.2.1 Auxiliary Assertions. 75
3.2.2 Imbedding of M(H? -+ Hlp) into M(H?~l - Lp). 76
3.2.3 Estimates for Derivatives of a Multiplier. 78
3.2.4 Multiplicative Inequality for the Strichartz Function . 79
3.2.5 Auxiliary Properties of the Bessel Kernel Gi. 80
3.2.6 Upper Bound for the Norm of a Multiplier. 81
3.2.7 Lower Bound for the Norm of a Multiplier. 85
3.2.8 Description of the Space M(H™ -? Hlp). 86
3.2.9 Equivalent Norm in M(H™ - Hlp) Involving
the Norm in I/mp/(m_/). 87
3.2.10 Characterization of M(H™ - Hlp), m I, Involving
the Norm in Li)Unif. 89
3.2.11 The Space M{H™ -»¦ Hlp) for mp n. 95
3.3 One-Sided Estimates for the Norm in M(#™ - Hlp). 95
3.3.1 Lower Estimate for the Norm in M(#™ - Hlp)
Involving Morrey Type Norms. 96
3.3.2 Upper Estimate for the Norm in M(H™ - Hlp)
Involving Marcinkiewicz Type Norms. 96
3.3.3 Upper Estimates for the Norm in M(#™ - Hl)
Involving Norms in Hln/m . 98
3.4 Upper Estimates for the Norm in Mill™ - Hlp) by Norms
in Besov Spaces. 99
3.4.1 Auxiliary Assertions. 99
3.4.2 Properties of the Space B% oo . 103
Contents VII
3.4.3 Estimates for the Norm in M{Hpn - Hlp)
by the Norm in B^ .108
3.4.4 Estimate for the Norm of a Multiplier in Jl/i/^R1)
by the (/-Variation.110
3.5 Miscellaneous Properties of Multipliers in M{H™ - H'p).Ill
3.6 Spectrum of Multipliers in Hlp and Hp,1 .115
3.6.1 Preliminary Information.115
3.6.2 Facts from Nonlinear Potential Theory.117
3.6.3 Main Theorem.118
3.6.4 Proof of Theorem 3.6.1.120
3.7 The Space M(/i™ - h!p).122
3.8 Positive Homogeneous Multipliers .125
3.8.1 The Space M{Hpn(dBx) - Hp{dB^)) .125
3.8.2 Other Normalizations of the Spaces h'pn and Hp".127
3.8.3 Positive Homogeneous Elements of the Spaces
M(h'» - h'p) and M(H? - Hlp).130
The Space M(B™ - Blp) with p 1.133
4.1 Introduction.133
4.2 Properties of Besov Spaces.134
4.2.1 Survey of Known Results.134
4.2.2 Properties of the Operators Dp,/ and Dpj .136
4.2.3 Pointwise Estimate for Bessel Potentials.138
4.3 Proof of Theorem 4.1.1.141
4.3.1 Estimate for the Product of First Differences.141
4.3.2 Trace Inequality for B*, p 1 .143
4.3.3 Auxiliary Assertions Concerning A/(B™ — Blp) .145
4.3.4 Lower Estimates for the Norm in M{B1pn -+ Blp).146
4.3.5 Proof of Necessity in Theorem 4.1.1.149
4.3.6 Proof of Sufficiency in Theorem 4.1.1 .155
4.3.7 The Case rap n .164
4.3.8 Lower and Upper Estimates for the Norm
in M(B™ - B'p).165
4.4 Sufficient Conditions for Inclusion into M(W™ — Wlp)
with Noninteger m and / .166
4.4.1 Conditions Involving the Space B£ x .166
4.4.2 Conditions Involving the Fourier Transform.168
4.4.3 Conditions Involving the Space Blqp .170
4.5 Conditions Involving the Space Hln, .173
4.6 Composition Operator on M{W™ - Wl).174
n/ri
•p ~^ "pi
VIII Contents
5 The Space M(B™ — B[).179
5.1 Trace Inequality for Functions in B\ (Rn).179
5.1.1 Auxiliary Facts .180
5.1.2 Main Result.183
5.2 Properties of Functions in the Space B\ (Kn).185
5.2.1 Trace and Imbedding Properties.185
5.2.2 Auxiliary Estimates for the Poisson Operator.189
5.3 Descriptions of M(Bf -+ B\) with Integer I .193
5.3.1 A Norm in M{B? - B[) .194
5.3.2 Description of M(B? - B[) Involving Du.199
5.3.3 M{B^{Wl) - B[{Rn)) as the Space of Traces.201
5.3.4 Interpolation Inequality for Multipliers.202
5.4 Description of the Space M{Bf - B[) with Noninteger I _203
5.5 Further Results on Multipliers in Besov and Other Function
Spaces . 206
5.5.1 Peetre's Imbedding Theorem.206
5.5.2 Related Results on Multipliers in Besov
and Triebel-Lizorkin Spaces.208
5.5.3 Multipliers in BMO.210
6 Maximal Algebras in Spaces of Multipliers .213
6.1 Introduction.213
6.2 Pointwise Interpolation Inequalities for Derivatives.214
6.2.1 Inequalities Involving Derivatives of Integer Order.214
6.2.2 Inequalities Involving Derivatives of Fractional Order .215
6.3 Maximal Banach Algebra in M(W™ - Wlp) .220
6.3.1 The Case p 1 .220
6.3.2 Maximal Banach Algebra in M(Wp -»¦ W[) .224
6.4 Maximal Algebra in Spaces of Bessel Potentials .227
6.4.1 Pointwise Inequalities Involving the Strichartz
Function.227
6.4.2 Banach Algebra A™'1.231
6.5 Imbeddings of Maximal Algebras.233
7 Essential Norm and Compactness of Multipliers.241
7.1 Auxiliary Assertions.243
7.2 Two-Sided Estimates for the Essential Norm. The Case
m I.248
7.2.1 Estimates Involving Cutoff Functions.248
7.2.2 Estimate Involving Capacity (The Case
mp n, p 1).250
7.2.3 Estimates Involving Capacity (The Case
mp = n, p 1).257
7.2.4 Proof of Theorem 7.0.3 .261
7.2.5 Sharpening of the Lower Bound for the Essential
Norm in the Case m /, mp n, p 1.262
Contents IX
7.2.6 Estimates of the Essential Norm for mp n, p 1
and for p = 1 .263
7.2.7 One-Sided Estimates for the Essential Norm.266
7.2.8 The Space of Compact Multipliers.267
7.3 Two-Sided Estimates for the Essential Norm
in the Case m = I .270
7.3.1 Estimate for the Maximum Modulus of a Multiplier
in Wp by its Essential Norm.270
7.3.2 Estimates for the Essential Norm Involving Cutoff
Functions (The Case Ip n, p 1).272
7.3.3 Estimates for the Essential Norm Involving Capacity
(The Case Ip n, p 1).277
7.3.4 Two-Sided Estimates for the Essential Norm
in the Cases Ip n, p 1, and p = \ .278
7.3.5 Essential Norm in MWlp.281
Traces and Extensions of Multipliers.285
8.1 Introduction.285
8.2 Multipliers in Pairs of Weighted Sobolev Spaces in W\_ .285
8.3 Characterization of M{W£(i - W°-a).288
8.4 Auxiliary Estimates for an Extension Operator.292
8.4.1 Pointwise Estimates for T-y and VT-y.292
8.4.2 Weighted Lp-Estimates for T-y and VT7 .294
8.5 Trace Theorem for the Space M(W^ -» W°-a) .297
8.5.1 The Case l \.298
8.5.2 The Case l \.301
8.5.3 Proof of Theorem 8.5.1 for I 1.303
8.6 Traces of Multipliers on the Smooth Boundary of a Domain. . . 304
8.7 MWj,(Rn) as the Space of Traces of Multipliers in the
Weighted Sobolev Space W£0{Rn+m).305
8.7.1 Preliminaries .305
8.7.2 A Property of Extension Operator.306
8.7.3 Trace and Extension Theorem for Multipliers.308
8.7.4 Extension of Multipliers from Rn to R"+1 .311
8.7.5 Application to the First Boundary Value Problem
in a Half-Space.311
8.8 Traces of Functions in i\/W^(Kn+m) on K" .312
8.8.1 Auxiliary Assertions.313
8.8.2 Trace and Extension Theorem .315
8.9 Multipliers in the Space of Bessel Potentials as Traces
of Multipliers.319
8.9.1 Bessel Potentials as Traces .319
8.9.2 An Auxiliary Estimate for the Extension Operator T . 320
8.9.3 MHlp as a Space of Traces.322
Contents
Sobolev Multipliers in a Domain, Multiplier Mappings
and Manifolds.325
9.1 Multipliers in a Special Lipschitz Domain.326
9.1.1 Special Lipschitz Domains.326
9.1.2 Auxiliary Assertions.326
9.1.3 Description of the Space of Multipliers.329
9.2 Extension of Multipliers to the Complement of a Special
Lipschitz Domain.332
9.3 Multipliers in a Bounded Domain.336
9.3.1 Domains with Boundary in the Class C01.336
9.3.2 Auxiliary Assertions.337
9.3.3 Description of Spaces of Multipliers in a Bounded
Domain with Boundary in the Class C0'1.339
9.3.4 Essential Norm and Compact Multipliers
in a Bounded Lipschitz Domain.340
9.3.5 The Space MLp(tt) for an Arbitrary
Bounded Domain.346
9.4 Change of Variables in Norms of Sobolev Spaces.350
9.4.1 (p, /)-Diffeomorphisms.350
9.4.2 More on (p, Z)-Diffeomorphisms.352
9.4.3 A Particular (p, Z)-Diffeomorphism.353
9.4.4 (p, O-Manifolds.356
9.4.5 Mappings T™1' of One Sobolev Space into Another_357
9.5 Implicit Function Theorems.364
9.6 The Space M{W™{Q) - Wlp{Q)) .367
9.6.1 Auxiliary Results.367
9.6.2 Description of the Space M{W™{fl) - Wp{H)).369
Part II Applications of Multipliers to Differential and Integral
Operators
10 Differential Operators in Pairs of Sobolev Spaces.373
10.1 The Norm of a Differential Operator: Wfi -* W^~k.373
10.1.1 Coefficients of Operators Mapping Wp into Wph~k
as Multipliers.374
10.1.2 A Counterexample.378
10.1.3 Operators with Coefficients Independent
of Some Variables.379
10.1.4 Differential Operators on a Domain.382
10.2 Essential Norm of a Differential Operator.384
10.3 Fredholm Property of the Schrodinger Operator.386
10.4 Domination of Differential Operators in E™.387
Contents XI
11 Schrodinger Operator and M(w* —? w^1) .391
11.1 Introduction.391
11.2 Characterization of M{w\ — w^1) and the Schrodinger
Operator on wi, .393
11.3 A Compactness Criterion.407
11.4 Characterization of M(Wj - W2l) .411
11.5 Characterization of the Space M(w\(Q) —* w^1^)).416
11.6 Second-Order Differential Operators Acting from w\ to w^1 . . 421
12 Relativistic Schrodinger Operator
and M(W21/2 - W^1"2).427
12.1 Auxiliary Assertions.427
12.1.1 Main Result.436
12.2 Corollaries of the Form Boundedness Criterion and Related
Results.441
13 Multipliers as Solutions to Elliptic Equations.445
13.1 The Dirichlet Problem for the Linear Second-Order Elliptic
Equation in the Space of Multipliers.445
13.2 Bounded Solutions of Linear Elliptic Equations
as Multipliers.447
13.2.1 Introduction.447
13.2.2 The Case /? 1.448
13.2.3 The Case /3 = 1.452
13.2.4 Solutions as Multipliers from W^w{p){Q) into W^O) .454
13.3 Solvability of Quasilinear Elliptic Equations in Spaces
of Multipliers.456
13.3.1 Scalar Equations in Divergence Form .457
13.3.2 Systems in Divergence Form.458
13.3.3 Dirichlet Problem for Quasilinear Equations
in Divergence Form.461
13.3.4 Dirichlet Problem for Quasilinear Equations
in Nondivergence Form.463
13.4 Coercive Estimates for Solutions of Elliptic equations
in Spaces of Multipliers .467
13.4.1 The Case of Operators in Rn .467
13.4.2 Boundary Value Problem in a Half-Space.469
13.4.3 On the Loo-Norm in the Coercive Estimate.473
13.5 Smoothness of Solutions to Higher Order Elliptic Semilinear
Systems.474
13.5.1 Composition Operator in Classes of Multipliers .474
13.5.2 Improvement of Smoothness of Solutions to Elliptic
Semilinear Systems.477
XII Contents
14 Regularity of the Boundary in Lp-Theory of Elliptic
Boundary Value Problems.479
14.1 Description of Results.479
14.2 Change of Variables in Differential Operators.481
14.3 Fredholm Property of the Elliptic Boundary Value Problem . . . 483
14.3.1 Boundaries in the Classes Mp~1/p, Wp~1/p,
and Mlp'l/p{5).483
14.3.2 A Priori Lp-Estimate for Solutions and Other
Properties of the Elliptic Boundary Value Problem . 484
14.4 Auxiliary Assertions.489
14.4.1 Some Properties of the Operator T.489
14.4.2 Properties of the Mappings A and x.490
14.4.3 Invariance of the Space Wlp n Wp Under a Change
of Variables.492
14.4.4 The Space W~k for a Special Lipschitz Domain.496
14.4.5 Auxiliary Assertions on Differential Operators
in Divergence Form.498
14.5 Solvability of the Dirichlet Problem in Wlp(Q).502
14.5.1 Generalized Formulation of the Dirichlet Problem.502
14.5.2 A Priori Estimate for Solutions of the Generalized
Dirichlet Problem.502
14.5.3 Solvability of the Generalized Dirichlet Problem.503
14.5.4 The Dirichlet Problem Formulated in Terms of Traces. . 504
14.6 Necessity of Assumptions on the Domain.507
14.6.1 A Domain Whose Boundary is in M%/2 n C1
but does not Belong to M%/2(6).507
14.6.2 Necessary Conditions for Solvability of the Dirichlet
Problem.509
14.6.3 Boundaries of the Class Mp~1/p(S).510
14.7 Local Characterization of Mp~1/p( 5).513
14.7.1 Estimates for a Cutoff Function.513
14.7.2 Description of Mp~1/p(6) Involving a Cutoff Function . . 515
14.7.3 Estimate for si.516
14.7.4 Estimate for s2.520
14.7.5 Estimate for s3.523
15 Multipliers in the Classical Layer Potential Theory
for Lipschitz Domains .531
15.1 Introduction.531
15.2 Solvability of Boundary Value Problems in Weighted Sobolev
Spaces.537
15.2.1 (p, k, a)-Diffeomorphisms.537
15.2.2 Weak Solvability of the Dirichlet Problem.539
15.2.3 Main Result.542
Contents XIII
15.3 Continuity Properties of Boundary Integral Operators.547
15.4 Proof of Theorems 15.1.1 and 15.1.2.551
15.4.1 Proof of Theorem 15.1.1.551
15.4.2 Proof of Theorem 15.1.2.557
15.5 Properties of Surfaces in the Class Mep{5).559
15.6 Sharpness of Conditions Imposed on 0(2.562
15.6.1 Necessity of the Inclusion 0(2 € W^
in Theorem 15.2.1.562
15.6.2 Sharpness of the Condition 0(2 E B^ p.563
15.6.3 Sharpness of the Condition 0(2 G Mp(5)
in Theorem 15.2.1.564
15.6.4 Sharpness of the Condition 0(2 £ Mp{S)
in Theorem 15.1.1.566
15.7 Extension to Boundary Integral Equations of Elasticity.568
16 Applications of Multipliers to the Theory of Integral
Operators .573
16.1 Convolution Operator in Weighted Z/2-Spaces.573
16.2 Calculus of Singular Integral Operators with Symbols
in Spaces of Multipliers .575
16.3 Continuity in Sobolev Spaces of Singular Integral Operators
with Symbols Depending oni .579
16.3.1 Function Spaces.580
16.3.2 Description of the Space M(Hm-" - H'*) .582
16.3.3 Main Result .585
16.3.4 Corollaries.588
References.591
List of Symbols.605
Author and Subject Index.607 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Mazʹja, Vladimir Gilelevič 1937- Šapošnikova, T. O. 1946- |
| author_GND | (DE-588)121490602 (DE-588)13670493X |
| author_facet | Mazʹja, Vladimir Gilelevič 1937- Šapošnikova, T. O. 1946- |
| author_role | aut aut |
| author_sort | Mazʹja, Vladimir Gilelevič 1937- |
| author_variant | v g m vg vgm t o š to toš |
| building | Verbundindex |
| bvnumber | BV035103260 |
| classification_rvk | SK 600 SK 620 |
| ctrlnum | (OCoLC)262282914 (DE-599)DNB989198820 |
| dewey-full | 515.782 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.782 |
| dewey-search | 515.782 |
| dewey-sort | 3515.782 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| id | DE-604.BV035103260 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T22:14:57Z |
| indexdate | 2024-07-09T21:22:17Z |
| institution | BVB |
| isbn | 9783540694908 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016771204 |
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| physical | XIII, 609 S. 235 mm x 155 mm |
| publishDate | 2009 |
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| publisher | Springer |
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| series | Grundlehren der mathematischen Wissenschaften |
| series2 | Grundlehren der mathematischen Wissenschaften |
| spelling | Mazʹja, Vladimir Gilelevič 1937- Verfasser (DE-588)121490602 aut Theory of Sobolev multipliers with applications to differential and integral operators Vladimir G. Maz'ya ; Tatyana O. Shaposhnikova Berlin [u.a.] Springer 2009 XIII, 609 S. 235 mm x 155 mm txt rdacontent n rdamedia nc rdacarrier Grundlehren der mathematischen Wissenschaften 337 Multiplikator (DE-588)4040703-2 gnd rswk-swf Sobolev-Raum (DE-588)4055345-0 gnd rswk-swf Sobolev-Raum (DE-588)4055345-0 s Multiplikator (DE-588)4040703-2 s DE-604 Šapošnikova, T. O. 1946- Verfasser (DE-588)13670493X aut Erscheint auch als Online-Ausgabe 978-3-540-69492-2 Grundlehren der mathematischen Wissenschaften 337 (DE-604)BV000000395 337 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016771204&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Mazʹja, Vladimir Gilelevič 1937- Šapošnikova, T. O. 1946- Theory of Sobolev multipliers with applications to differential and integral operators Grundlehren der mathematischen Wissenschaften Multiplikator (DE-588)4040703-2 gnd Sobolev-Raum (DE-588)4055345-0 gnd |
| subject_GND | (DE-588)4040703-2 (DE-588)4055345-0 |
| title | Theory of Sobolev multipliers with applications to differential and integral operators |
| title_auth | Theory of Sobolev multipliers with applications to differential and integral operators |
| title_exact_search | Theory of Sobolev multipliers with applications to differential and integral operators |
| title_exact_search_txtP | Theory of Sobolev multipliers with applications to differential and integral operators |
| title_full | Theory of Sobolev multipliers with applications to differential and integral operators Vladimir G. Maz'ya ; Tatyana O. Shaposhnikova |
| title_fullStr | Theory of Sobolev multipliers with applications to differential and integral operators Vladimir G. Maz'ya ; Tatyana O. Shaposhnikova |
| title_full_unstemmed | Theory of Sobolev multipliers with applications to differential and integral operators Vladimir G. Maz'ya ; Tatyana O. Shaposhnikova |
| title_short | Theory of Sobolev multipliers |
| title_sort | theory of sobolev multipliers with applications to differential and integral operators |
| title_sub | with applications to differential and integral operators |
| topic | Multiplikator (DE-588)4040703-2 gnd Sobolev-Raum (DE-588)4055345-0 gnd |
| topic_facet | Multiplikator Sobolev-Raum |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016771204&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000395 |
| work_keys_str_mv | AT mazʹjavladimirgilelevic theoryofsobolevmultiplierswithapplicationstodifferentialandintegraloperators AT saposnikovato theoryofsobolevmultiplierswithapplicationstodifferentialandintegraloperators |