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 w1m and W1m ........ 7
1.1.1. The Case m = 1 ................................ 7
1.1.2. The Case m ≥ 1 ............................... 12
1.2. Trace Inequalities for Functions in wpm and Wpm,
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 l ≤ 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(W1m → W1l) ............ 35
2.3. Characterization of the Space M(Wpm - Wpl)
for p > 1 ............................................ 38
2.3.1. Another Characterization of the Space
M(Wpm - Wpl) for 0 < l < m, pm < n, p < 1 ..... 43
2.3.2. Characterization of the Space M(Wpm - Wpl)
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(Wpm(R+n) → Wpl(R+n)) ....................... 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(Wpm → Wpk) ................................ 54
2.6. The Space M(Wpm → Wql) ................................ 57
2.7. Certain Properties of Multipliers .................... 58
2.8. The Space M(wpm → wpl) ................................ 60
2.9. Multipliers in Spaces of Functions with
Bounded Variation .................................... 63
2.9.1. The Spaces M bv and M BV ..................... 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(Hpm → Hpl) ........................... 75
3.2.1. Auxiliary Assertions ......................... 75
3.2.2. Imbedding of M(Hpm → Hpl) into
M(Hpm-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
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(Hpm → Hpl) ......... 86
3.2.9. Equivalent Norm in M(Hpm → Hpl) Involving
the Norm in Lmp/(m-l) .......................... 87
3.2.10. Characterization of M(Hpn → Hpl), m > l,
Involving the Norm in L1,unif ................. 89
3.2.11. The Space M(Hpm → Hpl) for mp > n ............ 95
3.3. One-Sided Estimates for the Norm in M(Hpm → Hpl) ...... 95
3.3.1. Lower Estimate for the Norm in M(Hpm → Hpl)
Involving Morrey Type Norms .................. 96
3.3.2. Upper Estimate for the Norm in
M(Hpm → Hpl) Involving Marcinkiewicz
Type Norms ................................... 96
3.3.3. Upper Estimates for the Norm in
M(Hpm → Hpl) Involving Norms in Hn/ml .......... 98
3.4. Upper Estimates for the Norm in M(Hpm → Hpl) by
Norms in Besov Spaces ................................ 99
3.4.1. Auxiliary Assertions ......................... 99
3.4.2. Properties of the Space Bq,∞η ................ 103
3.4.3. Estimates for the Norm in M(Hpm → Hpl) by
the Norm in Bq,∞η ........................... 108
3.4.4. Estimate for the Norm of a Multiplier in
МНрl(R1) by the q-Variation .................. 110
3.5. Miscellaneous Properties of Multipliers in
M(Hpm → Hpl) ......................................... 111
3.6. Spectrum of Multipliers in Hpl and Hp'-l ............. 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(hpm → hpl) ............................... 122
3.8. Positive Homogeneous Multipliers .................... 125
3.8.1. The Space M(Hpm( ) - Hpl()) ............. 125
3.8.2. Other Normalizations of the Spaces hpm
and Hpm ...................................... 127
3.8.3. Positive Homogeneous Elements of the
Spaces M(hpm → hpl) and M(Hpm → Hpl) .......... 130
4. The Space M(Bpm → Bpl) 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  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 Bpk, p → 1 ............. 143
4.3.3. Auxiliary Assertions Concerning
M(Bpm → Bpl) ................................. 145
4.3.4. Lower Estimates for the Norm in
M(Bpm → Bpl) ................................. 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(Bpm → Bpl) .............................. 165
4.4. Sufficient Conditions for Inclusion into
M(Wpm → Wpl) with Noninteger m and l ................. 166
4.4.1. Conditions Involving the Space Вq,∞μ ......... 166
4.4.2. Conditions Involving the Fourier
Transform ................................... 168
4.4.3. Conditions Involving the Space Bqpl .......... 170
4.5. Conditions Involving the Space Hn/ml ................. 173
4.6. Composition Operator on M(Wpm → Wpl) ................. 174
5. The Space M(B1m → B1l) ..................................... 179
5.1. Trace Inequality for Functions in B1l(Rn) ............ 179
5.1.1. Auxiliary Facts ............................. 180
5.1.2. Main Result ................................. 183
5.2. Properties of Functions in the Space B1k(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(B1m → B1l) with Integer l .......... 193
5.3.1. A Norm in M(B1m → B1l) ....................... 194
5.3.2. Description of M(B1m → B1l) Involving  ... 199
5.3.3. M(B1m(Rn) → B1l(Rn)) as the Space
of Traces ................................... 201
5.3.4. Interpolation Inequality for Multipliers .... 202
5.4. Description of the Space M(B1m - B1l) 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(Wpm - Wpl) ............... 220
6.3.1. The Case p > 1 .............................. 220
6.3.2. Maximal Banach Algebra in M(W1m - W1l) ....... 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 Apm,l ......................... 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 > l) .............................. 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 > l ............................... 262
7.2.6. Estimates of the Essential Norm for
mр > 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 Wpl 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 MWpl ...................... 281
8. Traces and Extensions of Multipliers ...................... 285
8.1. Introduction ........................................ 285
8.2. Multipliers in Pairs of Weighted Sobolev Spaces
in R+n .............................................. 285
8.3. Characterization of M(Wpt,β → Wps,α) ................. 288
8.4. Auxiliary Estimates for an Extension Operator ....... 292
8.4.1. Pointwise Estimates for Tγ and ÑTγ .......... 292
8.4.2. Weighted Lp-Estimates for Tγ and ÑTγ ........ 294
8.5. Trace Theorem for the Space M(Wpt,β → Wps,α) .......... 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. MWpl(Rn) as the Space of Traces of Multipliers in
the Weighted Sobolev Space Wb,βk(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+n+1 .... 311
8.7.5. Application to the First Boundary Value
Problem in a Half-Space ..................... 311
8.8. Traces of Functions in MWpl(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. MHlp 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 MLp1(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 Tpm,l of One Sobolev Space into
Another ..................................... 357
9.5. Implicit Function Theorems .......................... 364
9.6. The Space  ............................ 367
9.6.1. Auxiliary Results ........................... 367
9.6.2. Description of the Space ..... 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: Wph → Wph-k .... 373
10.1.1. Coefficients of Operators Mapping Wph
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. Predholm Property of the Schrцdinger Operator ...... 386
10.4. Domination of Differential Operators in Rn ......... 387
11. Schrödinger Operator and M(w21 → w2-1) .................... 391
11.1. Introduction ....................................... 391
11.2. Characterization of M(w21 → w2-1) and the
Schrödinger Operator оn w21 ........................ 393
11.3. A Compactness Criterion ............................ 407
11.4. Characterization of M(W21 → W2-1) ................... 411
11.5. Characterization of the Space M( ...... 416
11.6. Second-Order Differential Operators Acting from
w21 to w2-1 ......................................... 421
12. Relativistic Schrödinger Operator and
M(W21/2 → W2-1/2) .......................................... 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 β > 1 .............................. 448
13.2.3. The Case β = 1 ............................. 452
13.2.4. Solutions as Multipliers from
 into W12,1(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 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
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 Mpl-1/p,
Wpl-1/p, and Mpl-1/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 Τ ........... 489
14.4.2. Properties of the Mappings λ and χ .......... 490
14.4.3. Invariance of the Space  Under a
Change of Variables ......................... 492
14.4.4. The Space Wp-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 Wpl(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 M23/2 Ç C1
but does not Belong to M23/2(δ) .............. 507
14.6.2. Necessary Conditions for Solvability of
the Dirichlet Problem ....................... 509
14.6.3. Boundaries of the Class Mpl-1/p(δ) ........... 510
14.7. Local Characterization of Mpl-1/p(δ) ................. 513
14.7.1. Estimates for a Cutoff Function ............. 513
14.7.2. Description of Mpl-1/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 Mpl(δ) .......... 559
15.6. Sharpness of Conditions Imposed on ∂Ω ............... 562
15.6.1. Necessity of the Inclusion ∂Ω Î Wpl in
Theorem 15.2.1 .............................. 562
15.6.2. Sharpness of the Condition ∂Ω Î B∞,pl ........ 563
15.6.3. Sharpness of the Condition ∂Ω Î Mpl(δ)
in Theorem 15.2.1 ........................... 564
15.6.4. Sharpness of the Condition ∂Ω Î Mpl(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
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