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Introduction 1

Part Ⅰ Description and Properties of Multipliers 7

1 Trace Inequalities for Functions in Sobolev Spaces 7

1.1 Trace Inequalities for Functions in wm 1 and Wm 1 7

1.1.1 The Case m1＝ 7

1.1.2 The Case m≥1 12

1.2 Trace Inequalities for Functions in wm p and Wm p,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 Lq-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（Wm 1→Wl 1） 35

2.3 Characterization of the Space M(Wm p→Wl p)for p＞1 38

2.3.1 Another Characterization of the Space M(Wm p→Wl p)for 0＜l＜m,pm≤n,p＞1 43

2.3.2 Characterization of the Space M(Wm p→Wl p)for pm＞n,p＞1 47

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(Wm p(Rn +)→Wl p(Rn +)) 50

2.4.1 Extension from a Half-Space 50

2.4.2 The Case p＞1 51

2.4.3 The Case p=1 53

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 Multipliers 58

2.8 The Space M(wm p→wl p) 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 Beasel Potential Spaces 70

3.1.2 Properties of the(p,m)-Capacity 71

3.1.3 Main Result 73

3.2 Description of M(Hm p→Hl p) 75

3.2.1 Auxiliary Assertions 75

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 Multiplier 78

3.2.4 Multiplicative Inequality for the Strichartz Function 79

3.2.5 Auxiliary Properties of the Bessel Kernel Gl 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(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,unif 89

3.2.11 The Space M(Hm p→Hl p)for mp＞n 95

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 Norms 96

3.3.2 Upper Estimate for the Norm in M(Hm p→Hl p)Involving Marcinkiewicz Type Norms 96

3.3.3 Upper Estimates for the Norm in M(Hm p→Hl p)Involving Norms in Hl n/m 98

3.4 Upper Estimates for the Norm in M(Hm p→Hl p)by Norms in Besov Spaces 99

3.4.1 Auxiliary Assertions 99

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-Variation 110

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 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(hm p→hl p ) 122

3.8 Positive Homogeneous Multipliers 125

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 p 127

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＞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 ？p,l and Dp,l 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 Bk,p,p＞1 143

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 4.1.1 149

4.3.6 Proof of Sufficiency in Theorem 4.1.1 155

4.3.7 The Case mp＞n 164

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 l 166

4.4.1 Conditions Involving the Space Bμ q，∞ 166

4.4.2 Conditions Involving the Fourier Transform 168

4.4.3 Conditions Involving the Space Bl q,p 170

4.5 Conditions Involving the Space Hl n/m 173

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 Facts 180

5.1.2 Maln Result 183

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

5.2.1 Trace and Imbedding Properties 185

5.2.2 Auxiliary Estimates for the Poisson Operator 189

5.3 Descriptions of M(Bm 1→Bl 1)with Integer l 193

5.3.1 A Norm in M(Bm 1→Bl 1) 194

5.3.2 Description of M(Bm 1→Bl 1)Involving D1,l 199

5.3.3 M(Bm 1(Rn)→Bl 1(Rn))as the Space of Traces 201

5.3.4 Interpolation Inequality for Multipliers 202

5.4 Description of the Space M(Bm 1→Bl 1)with Noninteger l 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(Wm p→Wl p) 220

6.3.1 The Case p＞1 220

6.3.2 Maximal Banach Algebra in M(Wm 1→Wl 1) 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 Am,l p 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＞l 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＞l,mp ≤n,p＞1 262

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=l 270

7.3.1 Estimate for the Maximum Modulus of a Multiplier in Wl p by its Essential Norm 270

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=1 278

7.3.5 Essential Norm in ？Wl p 281

8 Traces and Extensions of Multipliers 285

8.1 Introduction 285

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 Operator 292

8.4.1 Pointwise Estimates for Tγ and ？Tγ 292

8.4.2 Weighted Lp-Estimates for Tγ and ？Tγ 294

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

8.5.1 The Case l＜1 298

8.5.2 The Case l＞1 301

8.5.3 Proof of Theorem 8.5.1 for l＞1 303

8.6 Traces of Multipliers on the Smooth Boundary of a Domain 304

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 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 Rn+1 + 311

8.7.5 Application to the First Boundary Value Problem in a Half-Space 311

8.8 Traces of Functions in MWl p(Rn+m)on Rn 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 MHl p as a Space of Traces 322

9 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 C0,1 336

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 ML1 p（Ω）for an Arbitrary Bounded Domain 346

9.4 Change of Variables in Norms of Sobolev Spaces 350

9.4.1 (p,l)-Diffeomorphisms 350

9.4.2 More on(p,l)-Diffeomorphisms 352

9.4.3 A Particular(p,l)-Diffeomorphism 353

9.4.4 (p,l)-Manifolds 356

9.4.5 Mappings Tm,l p of One Sobolev Space into Another 357

9.5 Implicit Function Theorems 364

9.6 The Space M（？m p（Ω）→Wl p（Ω）） 367

9.6.1 Auxiliary Results 367

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

Part Ⅱ Applications of Multipliers to Differential and Integral Operators 369

10 Differential Operators in Pairs of Sobolev Spaces 373

10.1 The Norm of a Differential Operator:Wh p→Wh-k p 373

10.1.1 Coefficients of Operators Mapping Wh p into Wh-k p 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 Schr？dinger Operator 386

10.4 Domination of Differential Operators in Rn 387

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

11.1 Introduction 391

11.2 Characterization of M(w1 2→w-1 2)and the Schr？dinger Operator on w1 2 393

11.3 A Compactness Criterion 407

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 2 421

12 Relativistic Schr？dinger Operator and M(W1/2 2→W-1/2 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 β＞1 448

13.2.3 The Case β＝1 452

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 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 L∞-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

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 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 Problem 484

14.4 Auxiliary Assertions 489

14.4.1 Some Properties of the Operator T 489

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 Variables 492

14.4.4 The Space W-k p 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 Wl p（Ω） 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 M3/2 2 ∩ C1 but does not Belong to M3/2 2（δ） 507

14.6.2 Necessary Conditions for Solvability of the Dirichlet Problem 509

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 Function 513

14.7.2 Description of Ml-1/p p（δ）Involving a Cutoff Function 515

14.7.3 Estimate for s1 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，α）-Diffeomorphisms 537

15.2.2 Weak Solvability of the Dirichlet Problem 539

15.2.3 Main Result 542

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 Me p（δ） 559

15.6 Sharpness of Conditions Imposed on ？Ω 562

15.6.1 Necessity of the Inclusion ？Ω ∈We p in Theorem 15.21 562

15.6.2 Sharpness of the Condition ？Ω ∈Be ∞,p 563

15.6.3 Sharpness of the Condition ？Ω ∈Me p（δ） in Theorem 15.2.1 564

15.6.4 Sharpness of the Condition ？Ω ∈Me p（δ） 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 L2-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 on x 579

16.3.1 Function Spaces 580

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

16.3.3 Main Result 585

16.3.4 Corollaries 588

References 591

List of Symbols 605

Author and Subject Index 607