Interpolation Theory Function Spaces Differential Operators

Interpolation Theory Function Spaces Differential Operators

By Hans Triebel

2nd revised and enlarged edition

Johann Ambrosius Barth Verlag Heidelberg • Leipzig

CONTENTS

1. Interpolation Theory in Banacb Spaces 16

1.1. Introduction 15 1.1.1. Abstract Interpolation Theory 15 1.1.2. Concrete Interpolation Theorems 16 1.1.3. Remarks on the Structure of the First Chapter 17

1.2. General Interpolation Methods 18 1.2.1. Interpolation Couples 18 1.2.2. Interpolation Functors 19 1.2.3. A0 r\ Ax and A„ + At as Interpolation Spaces 22 1.2.4. Retractions and Coretractions 22

1.3. TheK-Method 23 1.3.1. The ^-Functional 23 1.3.2. The Spaces (A0, Аг)вуд 24 1.3.3. Properties of the Spaces (A0, Аг)^ч 25

1.4. The L-Method 28 1.4.1. The K*- and the i*-Functional 28 1.4.2. Equivalence of the K- and the L-Method 29

1.5. The Mean-Methods 30 1.5.1. Preliminaries 31 1.5.2. First Equivalence Theorem 33 1.5.3. Second Equivalence Theorem 35

1.6. TheJ-Method 38 1.6.1. Equivalence Theorem 38

1.6.2. Density of A0 r^ Ax in (A0, Аг)^р 39

1.7. Discrete Methods 40

1.8. The Trace Method 41 1.8.1. The Spaces Vm(p0,r]0, A0; р^щ, At) and T™(p0, щ, А0; р1г %, А ) • • • • 42 1.8.2. Equivalence Theorem 44 1.8.3. Embedding Theorem 49 1.8.4. Quasilinearizable Interpolation Couples 51 1.8.5. Generalization of Embedding Theorem 1.8.3 53 1.9. Complex Methods 55 1.9.1. The Spaces F(A0,A1,y) and F_(A0,A1,y) 56 1.9.2. The Spaces [A0, AJe, [A0, AJe,? and [A^yA^s,^. 58 1.9.3. Properties of the Spaces [A0, A-jg 59

8 Contents

1.10. Reiteration Theorem 61 1.10.1. The Classes К(в, A0, Аг) and J(ß, A0, AJ 61 1.10.2. Reiteration Theorem 62 1.10.3. The Complex Method and the Reiteration Theorem 64

1.11. Duality Theory 68 1.11.1. The Dual Space of lp(A) 68 1.11.2. Duality Theory for the Real Method 69 1.11.3. Duality Theory for the Complex Method 71

1.12. Interpolation Theory for Quasilinearizable Interpolation Couples 72 1.12.1. A General Interpolation Theorem 73 1.12.2. Generalization of the Interpolation Theorem 1.12.1 74

1.13. Semi-Groups of Operators and Interpolation Spaces 75 1.13.1. Semi-Groups of Operators 75 1.13.2. The Spaces (A, D{Am))e<P [Part I] 76 1.13.3. The Spaces Km . . .' 81 1.13.4. Properties of the Spaces Km 83 1.13.5. The Spaces (A, D(Am))e,p [Part II] 86 1.13.6. The Spaces {A, Km)e,p ' 87

1.14. Positive Operators and Interpolation Spaces 91 1.14.1. Positive Operators 91 1.14.2. The Spaces (A, D{Am))9t „ 92 1.14.3. Equivalent Norms in the Spaces (A,D(Am))etP 93

1.14.4. The Spaces (А, о D(Ap))e>p 94

1.14.5. Analytic Semi-Groups and Interpolation Spaces 95

1.15. Fractional Powers of Positive Operators and Interpolation Spaces 98 1.15.1. Fractional Powers of Positive Operators 98 1.15.2. Properties of Fractional Powers of Positive Operators 100 1.15.3. Domains of Definition of Fractional Powers of Positive Operators 103 1.15.4. Reiteration Theorems 105

1.15.5. The Spaces (А, о D{A?))e>p 106

1.16. Interpolation Properties of Entropy Ideals and Width Ideals 107 1.16.1. Entropy Ideals and Width Ideals 107 1.16.2. Interpolation Properties of Entropy Ideals 112 1.16.3. Interpolation Properties of Width Ideals 115 1.16.4. Interpolation Properties of Compact Operators 117

1.17. Interpolation of Subspaces and Factor Spaces 118 1.17.1. Interpolation of Subspaces 118 1.17.2. Interpolation of Factor Spaces 119

1.18. Examples and Applications 120 1.18.1. Interpolation of the Spaces lp(Aj) 120 1.18.2. Interpolation of the Spaces 1%(A) 123 1.18.3. Interpolation of the Spaces lp(A) 125 1.18.4. Interpolation of the Spaces Lp(A) 127 1.18.5. Interpolation of the Spaces LPtU{x)(A) 130

Contents 9

1.18.6. Interpolation of the Spaces Lp(A). Lorentz Spaces 131 1.18.7. The Classical Interpolation Theorems 136 1.18.8. The Theorem of F . HATJSDOKFF and W. H. YOUNG 138 1.18.9. Convolution Integrals in Rn 139 1.18.10. Self-Adjoint Operators and Interpolation Theory 141

1.19. Complements 143 1.19.1. Further Interpolation Methods in Banach Spaces 143 1.19.2. Interpolation Functions 144 1.19.3. Interpolation Spaces in {-Lj, Xoo} and in General Interpolation Couples . . . 144 1.19.4. Interpolation Scales 145 1.19.6. Interpolation Properties of Bilinear Forms 145 1.19.6. Abstract Embedding Theorems for Interpolation Spaces 146 1.19.7. Interpolation Theory for Norm Ideals in Hilbert Spaces 146 1.19.8. Interpolation Theory for Quasi-Norm Ideals in Banach Spaces 147 1.19.9. Non-Commutative Interpolation 148 1.19.10. Interpolation-ra-Tuples 149 1.19.11. Interpolation Theory in General Spaces, Non-Linear Interpolation Theory . 149 1.19.12. Applications 149

2. Lebesgue-Besov Spaces without Weights in i?„ and R% 161

2.1. Introduction 151 2.2. Integral Operators and Fourier Multipliers 152 2.2.1. Distributions 152 2.2.2. Mapping Properties of Integral Operators 153 2.2.3. Singular Integral Operators 157 2.2.4. Multiplier Theorem 161

2.3. The Spaces B'M(Rn), FpJRn), Hp(Rn), and Wr(Rn) 168 2.3.1. Definitions , 168 2.3.2. The Spaces BpJRn) and F'r,,(Rn) 172 2.3.3. The Spaces H P \ R „ ) . . . .' 177 2.3.4. Lift Property 180

2.4. Interpolation Theory for the Spaces B'p,,(Rn) and F'M(Rn) 181 2.4.1. Interpolation of the Spaces B'Pi,(Rn) 181 2.4.2. Interpolation of the Spaces F'p[,(Rn) 184

2.5. Equivalent Norms in the Spaces B'Pi,(Rn) 187 2.5.1. Equivalent Norms and Translation Groups 187 2.5.2. Equivalent Norms and Gauss-Weierstrass Semi-Groups 190 2.5.3. Equivalent Norms and Cauchy-Poisson Semi-Groups 192 2.5.4. Equivalent Norms and Approximation 196

2.6. Duality Theory for the Spaces Bp,,{Rn) and F'p,,(Rn) 198 2.6.1. The Dual Spaces of B'pJRn) and H'p(Rn) 198 2.6.2. The Dual Spaces of F'M(Rn) 199

2.7. The Holder Spaces С"(Д„) 200 2.7.1. Definition of the Holder Spaces 200 2.7.2. Interpolation and Equivalent Norms 201

2.8. Embedding Theorems for Different Metrics 202 2.8.1. Embedding Theorem 203 2.8.2. Other Proofs of Theorem 2.8.1(a) 207

10 Contents

2.9. Direct and Inverse Embedding Theorems (Embedding on the Boundary) . . 209 2.9.1. Direct and Inverse Embedding Theorems for the Sobolev Spaces (I = n — 1) . 210 2.9.2. The Spaces 1 Р £ Ы . ( Д Я ) and И7>1п.(Д+) 2 1 3

2.9.3. Direct and Inverse Embedding Theorems for the Lebesgue-Besov Spaces (2 = n - 1) 218

2.9.4. Direct and Inverse Embedding Theorems for Lebesgue-Besov Spaces (General Case) 223

2.10. The Spaces H'p(R+n) and B*Piq(B$) 226

2.10.1. Interpolation of the Spaces H'P(R+) and B'P<„(B+) 226 2.10.2. Duality Theory [Part I ] 227 2.10.3. Lift Property 231 2.10.4 Interpolation of the Spaces Hr(R+), H'P{R+), B'p,,{Ri), and B'p,,{Bi) . . . . 233 2.10.5. Duality Theory [Part I I ] 234

2.11. Structure Theory 235 2.11.1. Preliminaries (Properties of the Spaces Lp and lp) 236 2.11.2. Structure of the Spaces H'P{BK) and B'M(Rn) 237

2.11.3. Structure of the Spaces HP(R + ) , H'P(Ri), B'Pil(Rt), and B'P,,(B+) 240

2.12. Diversity of the Spaces B'p,,(Rn) and H',(Rn) 241

2.13. Anisotropic Spaces 243 2.13.1. Definitions 243 2.13.2. Interpolation Theorem 244

3. Lebesgue-Besov Spaces with Weights in Domains 245

3.1. Introduction 245

3.2. Definitions and Fundamental Properties 245 3.2.1. The Spaces Wp(ß; a) 246 3.2.2. Properties of the Spaces WP(Q; a) 248 3.2.3. The Spaces B'Pi,{U; Q"; Q") and H'P(Q; Q"; Q") 250 3.2.4. Properties of the Spaces BP,,{Q; Q"; QV) and H'p(Q; Q?; Q") 252 3.2.5. Equivalent Norms and Compact Embedding in WP(Q) [Part I ] 258 3.2.6. The Spaces WP(Q;Q^; Q") with v < /г + sp 259

3.3. Interpolation Theory for the Spaces И''™(й; а) 266 3.3.1. The Spaces WP(Q; a) with Weight Functions of Type 1 266 3.3.2. The Spaces WP(Q; a) with Weight Functions of Type 2 268 3.3.3. The Spaces W"(Q; a) with Weight Functions of Type 3 270 3.3.4. The Spaces WP(Q; a) with Weight Functions of Type 4 273

3.4. Interpolation Theory for the Spaces B'p,g{Q; qf1; Q") and HP(Q;^; Q") . . . 273 3.4.1. Preparatory Lemma 273 3.4.2. Interpolation Theorem 275 3.4.3. Interpolation of the Spaces WP(Q; g''; Q") with v < /x + sp 277

3.5. Embedding Theorems for Different Metrics 278 3.5.1. The Spaces B'p,r(Q; Q^; QV) and H'P(Q; Q"; Q") 278 3.5.2. The Spaces WP(Q; Qf; QV) with v < /i + sp 279

Contents 11

3.6. Direct and Inverse Embedding Theorems (Embedding on the Boundary) . . 280

3.6.1. Direct and Inverse Embedding Theorems for the Spaces WP(Q; d«(x)) . . . 280 3.6.2. Direct and Inverse Embedding Theorems for the Spaces B'Tf„(R^; a£) . . . 282 3.6.3. Direct and Inverse Embedding Theorems for the Spaces BPi,(Q; dx(x)) . . . 285 3.6.4. Direct and Inverse Embedding Theorems for the Spaces W™(Rn; a),

W7(Ä+;«T),B;,,(Ä,;of),andBJ (,(Ä+;ff) 286

3.7. Structure Theory 289

3.8. Embedding Operators and Width Numbers 290

3.8.1. Equivalent Norms and Compact Embedding in WP(Q) [Part I I ] 291 3.8.2. Compact Embedding in W*(Q; Q"; е"+,"р) 293 3.8.3. Compact Embedding in W'V(Q; e"; e"+lp) 296

3.9. The Spaces wp,ß(Rn) 298

3.9.1. Definition and Equivalent Norms 299 3.9.2. Interpolation Theory 302 3.9.3. Direct and Inverse Embedding Theorems (Embedding on the Boundary) . . 302 3.9.4. Structure Theory 305

3.10. Complements 305

3.10.1. Spaces with Weights g(x) ~ d"{x) 306 3.10.2. Spaces with Mixed Norms 306 3.10.3. The Spaces of L. D. KUDRJAVCEV; Generalizations 307 3.10.4. The Spaces of I. A. KIPRIJANOV 307 3.10.5. The Spaces of A. KTJFNEE 308 3.10.6. Axiomatics of a Class of Spaces with Weights 308

4. Lebesgue-Besov Spaces without Weights in Domains 309


4.2. Definitions, Extension Theorems 309

4.2.1. Definitions 310 4.2.2. First Extension Method 310 4.2.3. Second Extension Method 312 4.2.4. Equivalent Norms in WP(Q) 315

4.3. Interpolation Theory 316

4.3.1. The Spaces BPJÜ) and H'r(Q) 317

4.3.2. The Spaces B'„JQ), B'P,,{Q), HV(Q), and H'P(Q) 317 4.3.3. The Spaces Bp,q,{Bj}(Q) and H'p,{Bj}(Q) 320

4.4. Equivalent Norms in Sobolev-Besov Spaces 321

4.4.1. Sobolev-Besov Spaces in Domains of Cone-Type 321 4.4.2. Sobolev-Besov Spaces in Bounded Domains 323

4.5. The Holder Spaces C\Q) 324

4.5.1. Definition and Extension Theorem 325 4.5.2. Interpolation and Equivalent Norms 325

4.6. Embedding Theorems for Different Metrics, Inclusion Properties 327

4.6.1. Embedding Theorems for Arbitrary Domains 327 4.6.2. Embedding Theorems for Bounded Domains 328

12 Contents

4.7. Direct and Inverse Embedding Theorems (Embedding on the Boundary) . . 329 4.7.1. Direct and Inverse Embedding Theorems (I = n — 1) 329 4.7.2. Direct and Inverse Embedding Theorems (General Case) 330

4.8. Duality Theory 331 4.8.1. The Dual Spaces of B'v,q(Q) and Щ(и) 332 4.8.2. The Dual Spaces of B'^JD) and Щ(и) 332

4.9. Structure Theory 333 4.9.1. Elliptic Boundary Value Problems 333 4.9.2. Scales 335 4.9.3. Isomorphism Properties 337 4.9.4. Schauder Bases 338

4.10. Qualitative Properties of Embedding Operators 344 4.10.1. Embedding Operators in Bounded Domains of Cone-Type

(Approximation Numbers, Width Numbers) 344 4.10.2. Embedding Operators in Bounded C°°-Domains (Approximation Numbers,

Width Numbers) 348 4.10.3. Embedding Operators in Bounded Domains (s-Entropy) 365

4.11. Complements 357 4.11.1. "Per iodic" Spaces 357 4.11.2. Spaces with Dominating Mixed Derivatives 358 4.11.3. Spaces of Abstract Functions 359 4.11.4. The Spaces of L. HORMANDEK and L. R. VOLEVIC, B. P. PANEJACH 359 4.11.5. Spaces with Variable Order of Differentiation 360

6. Regular Elliptic Differential Operators 361


5.2. Regular Elliptic Differential Operators 361 5.2.1. Definitions 361 5.2.2. Elliptic Operators 364 5.2.3. Regular Elliptic Problems in В + 366

5.3. A-Priori-Estimates 368 5.3.1. The Spaces H'/{B+) 368 5.3.2. A-Priori-Estimates [Part I. JfJ, constant coefficients, Dirichlet problem] . . 370 5.3.3. A-Priori-Estimates [Part I I . B+, constant coefficients, general boundary

problem] 373 5.3.4. A-Priori-Estimates [Part I I I . Bounded domain, variable coefficients, general

boundary problem] 375

5.4. iy-Theory in Sobolev Spaces 378 5.4.1. Smoothness Properties 378 5.4.2. Adjoint Operators (i2-Theory) 380 5.4.3. The Basic Theorem of Lp-Theory in Sobolev Spaces 382 5.4.4. The Operators Ap 383 5.4.5. The Operators 3I„ 386 5.4.6. Complements 388

5.5. Boundary Value Problems [Part I ] 389 5.5.1. Homogeneous Boundary Value Problems 389 5.5.2. Non-Homogeneous Boundary Value Problems 390

Contents 13

5.6. Distributions of Eigenvalues, Associated Eigenvectors, and Green Functions . 392

5.6.1. Distributions of Eigenvalues, and Associated Eigenvectors in Hilbert Spaces 392 5.6.2. Distributions of Eigenvalues of Self-Adjoint Elliptic Differential Operators . 395 5.6.3. Associated Eigenvectors of Elliptic Differential Operators 397 5.6.4. Green Functions of Elliptic Differential Operators 397

5.7. Boundary Value Problems [Part I I ] 401

5.7.1. Lebesgue-Besov Spaces without Weights 401 5.7.2. Sobolev-Besov Spaces with Weights 402 5.7.3. Holder Spaces 404

6. Strongly Degenerate Elliptic Differential Operators • . . . 405


6.2. Definitions and Preliminaries 405

6.2.1. Definitions 405 6.2.2. Powers of Strongly Degenerate Elliptic Differential Operators 407 6.2.3 Properties of the Spaces S#X)(Q) 408

6.3. A-Priori-Estimates 409

6.3.1. Equivalent Norms in the Spaces WJ(ß; ^ ; QV) 410 6.3.2. A-Priori-Estimates 410

6.4. L2-Theory for - Л + QV(X), V > 2 413 6.4.1. Self-Adjointness 414 6.4.2. Eigenfunctions 416 6.4.3. Domains of Definition of Fractional Powers, Isomorphic Mappings 417

6.5. i„-Theory 418 6.5.1. A-Priori-Estimates (Generalization of Theorem 6.3.2) 418 6.5.2. Isomorphism Theorems 420

6.6. Distributions of Eigenvalues, Associated Eigenvectors, and Green Functions . 422 6.6.1. Distributions of Eigenvalues and Domains of Definition of Fractional Powers 423 6.6.2. Associated Eigenvectors 425 6.6.3. Green Functions 427

7. Legendre and Tricomi Differential Operators 429


7.2. Definitions 430 7.2.1. Legendre Differential Operators 430 7.2.2. Tricomi Differential Operators 431

7.3. Inequalities, Equivalent Norms, and Isomorphic Mappings 433 7.3.1. Integral Inequalities [Part I ] 434 7.3.2. Properties of the Spaces W2((a,b),pct) 437 7.3.3. Mappings in W?((a, b),pa) 439

7.4. Self-Adjoint Legendre Differential Operators 441 7.4.1. Self-Adjoint and Positive-Definite Operators 441 7.4.2. The Minimal Operator Am,k 443

14 Contents

7.4.3. Domains of Definition of the Operators Ä'J?h, l = 0, 1,2, 445 7.4.4. The Spaces C%((a, b)) 446

7.6. Non-Self-Adjoint Legendre Differential Operators 449

7.5.1. Associated Eigenvectors of the Operators Bm-k 449 7.5.2. Isomorphic Mappings 450

7.6. Tricomi Differential Operators 452 7.6.1. Elliptic Differential Operators on Compact C00-Manifolds 452 7.6.2. Integral Inequalities [Part II] 454 7.6.3. Self-Adjoint Tricomi Differential Operators of First Type 457 7.6.4. Non-Self-Adjoint Tricomi Differential Operators of First Type 461 7.6.5. The Spaces Cfi(,Q) 461 7.6.6. Tricomi Differential Operators of Second Type 463

7.7. Domains of Definition of Fractional Powers 463 7.7.1. Legendre Differential Operators (m = к = 1) 464 7.7.2. Legendre Differential Operators (General Case) 467 7.7.3. Tricomi Differential Operators of First Type 468

7.8. Distributions of Eigenvalues, Green Functions 468 7.8.1. Legendre Differential Operators 468 7.8.2. Tricomi Differential Operators of First Type 470 7.8.3. Tricomi Differential Operators of Second Type 471

7.9. Complements 473 7.9.1. Boundary Value Problems for Degenerate Elliptic Differential Operators . . 473 7.9.2. Tricomi Differential Operators, Analytic Functions, and Functions of Gevrey

Classes 474 7.9.3. Further Types of Degenerate Elliptic Differential Equations 475

8. Nuclear Function Spaces 476


8.2. The Spaces Х>(Л°°) 476 8.2.1. Nuclear (#)-Spaces 476 8.2.2. The Structure of the Spaces D{AX) 477

8.3. Structure of Nuclear Function Spaces 480 8.3.1. General Structure Theorem 480 8.3.2. The Spaces <SeW(ß) and C?(ß) 481 8.3.3. The Spaces CJf,(ß) and C"(Q) 481

Bibliography 483

Appendix 519

Table of Symbols 523

Author Index 526

Subject Index 531

Documents

Interpolation Theory Function Spaces Differential Operators