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Introduction1

Part Ⅰ Description and Properties of Multipliers7

1 Trace Inequalities for Functions in Sobolev Spaces7

1.1 Trace Inequalities for Functions in wm 1 and Wm 17

1.1.1 The Case m1＝7

1.1.2 The Case m≥112

1.2 Trace Inequalities for Functions in wm p and Wm p,p＞114

1.2.1 Preliminaries14

1.2.2 The(p,m)-Capacity16

1.2.3 Estimate for the Integral of Capacity of a Set Bounded by a Level Surface19

1.2.4 Estimates for Constants in Trace Inequalities22

1.2.5 Other Criteria for the Trace Inequality(1.2.29) with p＞125

1.2.6 The Fefferman and Phong Sufficient Condition28

1.3 Estimate for the Lq-Norm with respect to an Arbitrary Measure29

1.3.1 The case 1≤p＜q30

1.3.2 The case q＜p≤n/m30

2 Multipliers in Pairs of Sobolev Spaces33

2.1 Introduction33

2.2 Characterization of the Space M（Wm 1→Wl 1）35

2.3 Characterization of the Space M(Wm p→Wl p)for p＞138

2.3.1 Another Characterization of the Space M(Wm p→Wl p)for 0＜l＜m,pm≤n,p＞143

2.3.2 Characterization of the Space M(Wm p→Wl p)for pm＞n,p＞147

2.3.3 One-Sided Estimates for Norms of Multipliers in the Case pm≤n48

2.3.4 Examples of Multipliers49

2.4 The Space M(Wm p(Rn +)→Wl p(Rn +))50

2.4.1 Extension from a Half-Space50

2.4.2 The Case p＞151

2.4.3 The Case p=153

2.5 The Space M(Wm p→W-k p)54

2.6 The Space M(Wm p→Wl q)57

2.7 Certain Properties of Multipliers58

2.8 The Space M(wm p→wl p)60

2.9 Multipliers in Spaces of Functions with Bounded Variation63

2.9.1 The Spaces Mbv and MBV66

3 Multipliers in Pairs of Potential Spaces69

3.1 Trace Inequality for Bessel and Riesz Potential Spaces69

3.1.1 Properties of Beasel Potential Spaces70

3.1.2 Properties of the(p,m)-Capacity71

3.1.3 Main Result73

3.2 Description of M(Hm p→Hl p)75

3.2.1 Auxiliary Assertions75

3.2.2 Imbedding of M(Hm p→Hl p)into M(Hm-l p→Lp)76

3.2.3 Estimates for Derivatives of a Multiplier78

3.2.4 Multiplicative Inequality for the Strichartz Function79

3.2.5 Auxiliary Properties of the Bessel Kernel Gl80

3.2.6 Upper Bound for the Norm of a Multiplier81

3.2.7 Lower Bound for the Norm of a Multiplier85

3.2.8 Description of the Space M(Hm p→Hl p)86

3.2.9 Equivalent Norm in M(Hm p→Hl p)Involving the Norm inLmp/(m-l)87

3.2.10 Characterization of M(Hm p→Hl p),m＞l,Involving the Norm in L1,unif89

3.2.11 The Space M(Hm p→Hl p)for mp＞n95

3.3 One-Sided Estimates for the Norm in M(Hm p→Hl p)95

3.3.1 Lower Estimate for the Norm in M(Hm p→Hl p)Involving Morrey Type Norms96

3.3.2 Upper Estimate for the Norm in M(Hm p→Hl p)Involving Marcinkiewicz Type Norms96

3.3.3 Upper Estimates for the Norm in M(Hm p→Hl p)Involving Norms in Hl n/m98

3.4 Upper Estimates for the Norm in M(Hm p→Hl p)by Norms in Besov Spaces99

3.4.1 Auxiliary Assertions99

3.4.2 Properties of the Space Bμ q，∞103

3.4.3 Estimates for the Norm in M(Hm p→Hl p)by the Norm in Bμ q，∞108

3.4.4 Estimate for the Norm of a Multiplier in M Hl p(R1)by the q-Variation110

3.5 Miscellaneous Properties of Multipliers in M(Hm p→Hl p)111

3.6 Spectrum of Multipliers in Hl p and H-l p'115

3.6.1 Preliminary Information115

3.6.2 Facts from Nonlinear Potential Theory117

3.6.3 Main Theorem118

3.6.4 Proof of Theorem 3.6.1120

3.7 The Space M(hm p→hl p )122

3.8 Positive Homogeneous Multipliers125

3.8.1 The Space M（Hm p（？β1）→Hl p（？β1））125

3.8.2 Other Normalizations of the Spaces hm p and Hm p127

3.8.3 Positive Homogeneous Elements of the Spaces M(hm p→hl p)and M(Hm p→Hl p)130

4 The Space M(Bm p→Bl p)with p＞1133

4.1 Introduction133

4.2 Properties of Besov Spaces134

4.2.1 Survey of Known Results134

4.2.2 Properties of the Operators ？p,l and Dp,l136

4.2.3 Pointwise Estimate for Bessel Potentials138

4.3 Proof of Theorem 4.1.1141

4.3.1 Estimate for the Product of First Differences141

4.3.2 Trace Inequality for Bk,p,p＞1143

4.3.3 Auxiliary Assertions Concerning M(Bm p→Bl p)145

4.3.4 Lower Estimates for the Norm in M(Bm p→Bl p)146

4.3.5 Proof of Necessity in Theorem 4.1.1149

4.3.6 Proof of Sufficiency in Theorem 4.1.1155

4.3.7 The Case mp＞n164

4.3.8 Lower and Upper Estimates for the Norm in M(Bm→Bl p)165

4.4 Sufficient Conditions for Inclusion into M(Wm p→Wl p)with Noninteger m and l166

4.4.1 Conditions Involving the Space Bμ q，∞166

4.4.2 Conditions Involving the Fourier Transform168

4.4.3 Conditions Involving the Space Bl q,p170

4.5 Conditions Involving the Space Hl n/m173

4.6 Composition Operator on M(Wm p→Wl p)174

5 The Space M(Bm 1→Bl 1)179

5.1 Trace Inequality for Functions in Bl 1(Rn)179

5.1.1 Auxiliary Facts180

5.1.2 Maln Result183

5.2 Properties of Functions in the Space Bk 1(Rn)185

5.2.1 Trace and Imbedding Properties185

5.2.2 Auxiliary Estimates for the Poisson Operator189

5.3 Descriptions of M(Bm 1→Bl 1)with Integer l193

5.3.1 A Norm in M(Bm 1→Bl 1)194

5.3.2 Description of M(Bm 1→Bl 1)Involving D1,l199

5.3.3 M(Bm 1(Rn)→Bl 1(Rn))as the Space of Traces201

5.3.4 Interpolation Inequality for Multipliers202

5.4 Description of the Space M(Bm 1→Bl 1)with Noninteger l203

5.5 Further Results on Multipliers in Besov and Other Function Spaces206

5.5.1 Peetre's Imbedding Theorem206

5.5.2 Related Results on Multipliers in Besov and Triebel-Lizorkin Spaces208

5.5.3 Multipliers in BMO210

6 Maximal Algebras in Spaces of Multipliers213

6.1 Introduction213

6.2 Pointwise Interpolation Inequalities for Derivatives214

6.2.1 Inequalities Involving Derivatives of Integer Order214

6.2.2 Inequalities Involving Derivatives of Fractional Order215

6.3 Maximal Banach Algebra in M(Wm p→Wl p)220

6.3.1 The Case p＞1220

6.3.2 Maximal Banach Algebra in M(Wm 1→Wl 1)224

6.4 Maximal Algebra in Spaces of Bessel Potentials227

6.4.1 Pointwise Inequalities Involving the Strichartz Function227

6.4.2 Banach Algebra Am,l p231

6.5 Imbeddings of Maximal Algebras233

7 Essential Norm and Compactness of Multipliers241

7.1 Auxiliary Assertions243

7.2 Two-Sided Estimates for the Essential Norm.The Case m＞l248

7.2.1 Estimates Involving Cutoff Functions248

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 7.0.3261

7.2.5 Sharpening of the Lower Bound for the Essential Norm in the Case m＞l,mp ≤n,p＞1262

7.2.6 Estimates of the Essential Norm for mp＞n,p＞1 and for p=1263

7.2.7 One-Sided Estimates for the Essential Norm266

7.2.8 The Space of Compact Multipliers267

7.3 Two-Sided Estimates for the Essential Norm in the Case m=l270

7.3.1 Estimate for the Maximum Modulus of a Multiplier in Wl p by its Essential Norm270

7.3.2 Estimates for the Essential Norm Involving Cutoff Functions(The Case lp≤n,p＞1)272

7.3.3 Estimates for the Essential Norm Involving Capacity(The Case lp≤n,p＞1)277

7.3.4 Two-Sided Estimates for the Essential Norm in the Cases lp＞n,p＞1,and p=1278

7.3.5 Essential Norm in ？Wl p281

8 Traces and Extensions of Multipliers285

8.1 Introduction285

8.2 Multipliers in Pairs of Weighted Sobolev Spaces in Rn +285

8.3 Characterization of M(Wt，β p→Ws，α p)288

8.4 Auxiliary Estimates for an Extension Operator292

8.4.1 Pointwise Estimates for Tγ and ？Tγ292

8.4.2 Weighted Lp-Estimates for Tγ and ？Tγ294

8.5 Trace Theorem for the Space M(Wt，β p→Ws，α p)297

8.5.1 The Case l＜1298

8.5.2 The Case l＞1301

8.5.3 Proof of Theorem 8.5.1 for l＞1303

8.6 Traces of Multipliers on the Smooth Boundary of a Domain304

8.7 MWl p(Rn)as the Space of Traces of Multipliers in the Weighted Sobolev Space Wk p，β（Rn+m）305

8.7.1 Preliminaries305

8.7.2 A Property of Extension Operator306

8.7.3 Trace and Extension Theorem for Multipliers308

8.7.4 Extension of Multipliers from Rn to Rn+1 +311

8.7.5 Application to the First Boundary Value Problem in a Half-Space311

8.8 Traces of Functions in MWl p(Rn+m)on Rn312

8.8.1 Auxiliary Assertions313

8.8.2 Trace and Extension Theorem315

8.9 Multipliers in the Space of Bessel Potentials as Traces of Multipliers319

8.9.1 Bessel Potentials as Traces319

8.9.2 An Auxiliary Estimate for the Extension Operator T320

8.9.3 MHl p as a Space of Traces322

9 Sobolev Multipliers in a Domain,Multiplier Mappings and Manifolds325

9.1 Multipliers in a Special Lipschitz Domain326

9.1.1 Special Lipschitz Domains326

9.1.2 Auxiliary Assertions326

9.1.3 Description of the Space of Multipliers329

9.2 Extension of Multipliers to the Complement of a Special Lipschitz Domain332

9.3 Multipliers in a Bounded Domain336

9.3.1 Domains with Boundary in the Class C0,1336

9.3.2 Auxiliary Assertions337

9.3.3 Description of Spaces of Multipliers in a Bounded Domain with Boundary in the Class C0,1339

9.3.4 Essential Norm and Compact Multipliers in a Bounded Lipschitz Domain340

9.3.5 The Space ML1 p（Ω）for an Arbitrary Bounded Domain346

9.4 Change of Variables in Norms of Sobolev Spaces350

9.4.1 (p,l)-Diffeomorphisms350

9.4.2 More on(p,l)-Diffeomorphisms352

9.4.3 A Particular(p,l)-Diffeomorphism353

9.4.4 (p,l)-Manifolds356

9.4.5 Mappings Tm,l p of One Sobolev Space into Another357

9.5 Implicit Function Theorems364

9.6 The Space M（？m p（Ω）→Wl p（Ω））367

9.6.1 Auxiliary Results367

9.6.2 Description of the Space M（？m p（Ω）→Wl p（Ω））369

Part Ⅱ Applications of Multipliers to Differential and Integral Operators369

10 Differential Operators in Pairs of Sobolev Spaces373

10.1 The Norm of a Differential Operator:Wh p→Wh-k p373

10.1.1 Coefficients of Operators Mapping Wh p into Wh-k p as Multipliers374

10.1.2 A Counterexample378

10.1.3 Operators with Coefficients Independent of Some Variables379

10.1.4 Differential Operators on a Domain382

10.2 Essential Norm of a Differential Operator384

10.3 Fredholm Property of the Schr？dinger Operator386

10.4 Domination of Differential Operators in Rn387

11 Schr？dinger Operator and M(w1 2→w-1 2)391

11.1 Introduction391

11.2 Characterization of M(w1 2→w-1 2)and the Schr？dinger Operator on w1 2393

11.3 A Compactness Criterion407

11.4 Characterization of M(W1 2→W-1 2)411

11.5 Characterization of the Space M（？1 2（Ω）→w-1 2（Ω））416

11.6 Second-Order Differential Operators Acting from w1 2 to w-1 2421

12 Relativistic Schr？dinger Operator and M(W1/2 2→W-1/2 2)427

12.1 Auxiliary Assertions427

12.1.1 Main Result436

12.2 Corollaries of the Form Boundedness Criterion and Related Results441

13 Multipliers as Solutions to Elliptic Equations445

13.1 The Dirichlet Problem for the Linear Second-Order…Elliptic Equation in the Space of Multipliers445

13.2 Bounded Solutions of Linear Elliptic Equations as Multipliers447

13.2.1 Introduction447

13.2.2 The Case β＞1448

13.2.3 The Case β＝1452

13.2.4 Solutions as Multipliers from W1 2,w（ρ）（Ω）into W1 2,1（Ω）454

13.3 Solvability of Quasilinear Elliptic Equations in Spaces of Multipliers456

13.3.1 Scalar Equations in Divergence Form457

13.3.2 Systems in Divergence Form458

13.3.3 Dirichlet Problem for Quasilinear Equations in Divergence Form461

13.3.4 Dirichlet Problem for Quasilinear Equations in Nondivergence Form463

13.4 Coercive Estimates for Solutions of Elliptic equations in Spaces of Multipliers467

13.4.1 The Case of Operators in Rn467

13.4.2 Boundary Value Problem in a Half-Space469

13.4.3 On the L∞-Norm in the Coercive Estimate473

13.5 Smoothness of Solutions to Higher Order Elliptic Semilinear Systems474

13.5.1 Composition Operator in Classes of Multipliers474

13.5.2 Improvement of Smoothness of Solutions to Elliptic Semilinear Systems477

14 Regularity of the Boundary in Lp-Theory of Elliptic Boundary Value Problems479

14.1 Description of Results479

14.2 Change of Variables in Differential Operators481

14.3 Fredholm Property of the Elliptic Boundary Value Problem483

14.3.1 Boundaries in the Classes Ml-1/p p,Wl-1/p,p and Ml-1/p p（δ）483

14.3.2 A Priori Lp-Estimate for Solutions and Other Properties of the Elliptic Boundary Value Problem484

14.4 Auxiliary Assertions489

14.4.1 Some Properties of the Operator T489

14.4.2 Properties of the Mappings λ and ？490

14.4.3 Invariance of the Space Wl p ∩ ？h p Under a Change of Variables492

14.4.4 The Space W-k p for a Special Lipschitz Domain496

14.4.5 Auxiliary Assertions on Differential Operators in Divergence Form498

14.5 Solvability of the Dirichlet Problem in Wl p（Ω）502

14.5.1 Generalized Formulation of the Dirichlet Problem502

14.5.2 A Priori Estimate for Solutions of the Generalized Dirichlet Problem502

14.5.3 Solvability of the Generalized Dirichlet Problem503

14.5.4 The Dirichlet Problem Formulated in Terms of Traces504

14.6 Necessity of Assumptions on the Domain507

14.6.1 A Domain Whose Boundary is in M3/2 2 ∩ C1 but does not Belong to M3/2 2（δ）507

14.6.2 Necessary Conditions for Solvability of the Dirichlet Problem509

14.6.3 Boundaries of the Class Ml-1/p p（δ）510

14.7 Local Characterization of Ml-1/p p（δ）513

14.7.1 Estimates for a Cutoff Function513

14.7.2 Description of Ml-1/p p（δ）Involving a Cutoff Function515

14.7.3 Estimate for s1516

14.7.4 Estimate for s2520

14.7.5 Estimate for s3523

15 Multipliers in the Classical Layer Potential Theory for Lipschitz Domains531

15.1 Introduction531

15.2 Solvability of Boundary Value Problems in Weighted Sobolev Spaces537

15.2.1 （p,k，α）-Diffeomorphisms537

15.2.2 Weak Solvability of the Dirichlet Problem539

15.2.3 Main Result542

15.3 Continuity Properties of Boundary Integral Operators547

15.4 Proof of Theorems 15.1.1 and 15.1.2551

15.4.1 Proof of Theorem 15.1.1551

15.4.2 Proof of Theorem 15.1.2557

15.5 Properties of Surfaces in the Class Me p（δ）559

15.6 Sharpness of Conditions Imposed on ？Ω562

15.6.1 Necessity of the Inclusion ？Ω ∈We p in Theorem 15.21562

15.6.2 Sharpness of the Condition ？Ω ∈Be ∞,p563

15.6.3 Sharpness of the Condition ？Ω ∈Me p（δ） in Theorem 15.2.1564

15.6.4 Sharpness of the Condition ？Ω ∈Me p（δ） in Theorem 15.1.1566

15.7 Extension to Boundary Integral Equations of Elasticity568

16 Applications of Multipliers to the Theory of Integral Operators573

16.1 Convolution Operator in Weighted L2-Spaces573

16.2 Calculus of Singular Integral Operators with Symbols in Spaces of Multipliers575

16.3 Continuity in Sobolev Spaces of Singular Integral Operators with Symbols Depending on x579

16.3.1 Function Spaces580

16.3.2 Description of the Space M（Hm,μ→Hl,μ）582

16.3.3 Main Result585

16.3.4 Corollaries588

References591

List of Symbols605

Author and Subject Index607