.

0 (hodnocen0 x )

BK

2nd., rev. a augm. ed.

Heidelberg : Springer, c2011

xxviii, 866 s. ; 24 cm

ISBN 978-3-642-15563-5 (váz.)

Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen : : a series of comprehensive studies in mathematics ; 342

Obsahuje bibliografické odkazy a rejstřík

000196434

Contents // 1 Basic Properties of Sobolev Spaces... 1 // 1.1 The Spaces Llp(f2), V‘(f2) and Wj,(ß)... 1 // 1.1.1 Notation... 1 // 1.1.2 Local Properties of Elements in the Space Llp((2)... 2 // 1.1.3 Absolute Continuity of Functions in Lp((2)... 4 // 1.1.4 Spaces W’(ß) and Ѵ}(П) ... 7 // 1.1.5 Approximation of Functions in Sobolev Spaces by Smooth Functions in Q ... 9 // 1.1.6 Approximation of Functions in Sobolev Spaces by Functions in C°°(ß)... 10 // 1.1.7 Transformation of Coordinates in Norms of Sobolev Spaces... 12 // 1.1.8 Domains Starshaped with Respect to a Ball ... 14 // 1.1.9 Domains of the Class C0,1 and Domains Having the // Cone Property... 15 // 1.1.10 Sobolev Integral Representation... 16 // 1.1.11 Generalized Poincare Inequality... 20 // 1.1.12 Completeness of Wp(fi) and Vp{Q)... 22 // 1.1.13 The Space Lp(i2) and Its Completeness... 22 // 1.1.14 Estimate of Intermediate Derivative and Spaces // W n) and Llp(i2)... 23 // 1.1.15 Duals of Sobolev Spaces... 24 // 1.1.16 Equivalent Norms in Wp(f2)... 26 // 1.1.17 Extension of Functions in Vp(f2) onto Kn... 26 // 1.1.18 Removable Sets for Sobolev Functions... 28 // 1.1.19 Comments to Sect. 1.1... 29 // 1.2 Facts from Set Theory and Function Theory... 32 // 1.2.1 Two Theorems on Coverings... 32 // 1.2.2 Theorem on Level Sets of a Smooth Function... 35 // ix // x Contents // 1.2.3 Representation of the Lebesgue Integral as a Riemann // Integral along a Halfaxis ... 37 // 1.2.4 Formula for the Integral of Modulus

of the Gradient ... 38 // 1.2.5 Comments to Sect. 1.2... 39 // 1.3 Some Inequalities for Functions of One Variable... 40 // 1.3.1 Basic Facts on Hardy-type Inequalities... 40 // 1.3.2 Two-weight Extensions of Hardy’s Type Inequality in // the Case p < q... 42 // 1.3.3 Two-Weight Extensions of Hardy’s Inequality in the // Case p > q ... 48 // 1.3.4 Hardy-Type Inequalities with Indefinite Weights 51 // 1.3.5 Three Inequalities for Functions on (0, oo)... 57 // 1.3.6 Estimates for Differentiable Nonnegative Functions of // One Variable ... 59 // 1.3.7 Comments to Sect. 1.3... 62 // 1.4 Embedding Theorems of Sobolev Type... 63 // 1.4.1 D.R. Adams’ Theorem on Riesz Potentials... 64 // 1.4.2 Estimate for the Norm in Lq(Rn,/i) by the Integral of // the Modulus of the Gradient... 67 // 1.4.3 Estimate for the Norm in Lq(Rn,p) by the Integral of // the Modulus of the Ith Order Gradient... 70 // 1.4.4 Corollaries of Previous Results... 72 // 1.4.5 Generalized Sobolev Theorem... 73 // 1.4.6 Compactness Theorems... 76 // 1.4.7 Multiplicative Inequalities... 79 // 1.4.8 Comments to Sect. 1.4... 83 // 1.5 More on Extension of Functions in Sobolev Spaces... 87 // 1.5.1 Survey of Results and Examples of Domains... 87 // 1.5.2 Domains in EVp which Are Not Quasidisks... 91 // 1.5.3 Extension with Zero Boundary Data... 94 // 1.5.4 Comments to Sect. 1.5... 96 // 1.6 Inequalities for Functions with Zero Incomplete Cauchy Data . 99 // 1.6.1 Integral Representation for Functions of One // Independent

Variable... 99 // 1.6.2 Integral Representation for Functions of Several // Variables with Zero Incomplete Cauchy Data...100 // 1.6.3 Embedding Theorems for Functions with Zero // Incomplete Cauchy Data...102 // 1.6.4 Necessity of the Condition I < 2k...105 // 1.7 Density of Bounded Functions in Sobolev Spaces...107 // 1.7.1 Lemma on Approximation of Functions in Lp(f2) 107 // 1.7.2 Functions with Bounded Gradients Are Not Always // Dense in Lp((2)...108 // 1.7.3 A Planar Bounded Domain for Which П L Q) // Is Not Dense in L\\(Q)...109 // Contents X і // 1.7.4 Density of Bounded Functions in Lp((2) for // Paraboloids in Kn...112 // 1.7.5 Comments to Sect. 1.7...117 // 1.8 Maximal Algebra in Wlp(ii)...117 // 1.8.1 Main Result...117 // 1.8.2 The Space П Ьж(й) Is Not Always a Banach // Algebra...120 // 1.8.3 Comments to Sect. 1.8...121 // 2 Inequalities for Functions Vanishing at the Boundary 123 // 2.1 Conditions for Validity of Integral Inequalities // (the Case p = 1)...124 // 2.1.1 Criterion Formulated in Terms of Arbitrary Admissible // Sets...124 // 2.1.2 Criterion Formulated in Terms of Balls for Í2 = R” 126 // 2.1.3 Inequality Involving the Norms in Lq(f2,p) and // Lr{Q, u) (Case p = 1)...127 // 2.1.4 Case q Є (0,1) ...127 // 2.1.5 Inequality (2.1.10) Containing Particular Measures ... 132 // 2.1.6 Power Weight Norm of the Gradient on the // Right-Hand Side...133 // 2.1.7 Inequalities of Hardy-Sobolev Type as Corollaries of // Theorem 2.1.6/1 ...138 // 2.1.8 Comments

to Sect. 2.1...140 // 2.2 (p, <ř)-Capacity...141 // 2.2.1 Definition and Properties of the (p, 4>)-Capacity...141 // 2.2.2 Expression for the (p, Ф)-Сарасіїу Containing an // Integral over Level Surfaces...144 // 2.2.3 Lower Estimates for the (p, <ř)-Capacity...146 // 2.2.4 p-Capacity of a Ball...148 // 2.2.5 (p, <P)-Capacity for p = 1 ...149 // 2.2.6 The Measure m„_i and 2-Capacity...149 // 2.2.7 Comments to Sect. 2.2...151 // 2.3 Conditions for Validity of Integral Inequalities // (the Case p>l)...152 // 2.3.1 The (p, Ф)-Сарасіїагу Inequality...152 // 2.3.2 Capacity Minimizing Function and Its Applications ... 156 // 2.3.3 Estimate for a Norm in a Birnbaum-Orlicz Space 157 // 2.3.4 Sobolev Type Inequality as Corollary of Theorem 2.3.3.160 // 2.3.5 Best Constant in the Sobolev Inequality (p > 1)...160 // 2.3.6 Multiplicative Inequality (the Case p > 1)...162 // 2.3.7 Estimate for the Norm in Lq(Q,p.) with q < p (First // Necessary and Sufficient Condition)...165 // 2.3.8 Estimate for the Norm in Lq((2,p.) with q < p (Second // Necessary and Sufficient Condition)...167 // xii Contents // 2.3.9 Inequality with the Norms in Lq(Q,ß) and Lr(Q,v) // (the Case p > 1) ...171 // 2.3.10 Estimate with a Charge a on the Left-Hand Side...173 // 2.3.11 Multiplicative Inequality with the Norms in L4(i2,p) // and Lr{Q, ľ) (Case p > 1)...174 // 2.3.12 On Nash and Moser Multiplicative Inequalities...176 // 2.3.13 Comments to Sect. 2.3...177 // 2.4 Continuity and Compactness

of Embedding Operators of // Lp(i2) and Wp(Q) into Birnbaum-Orlicz Spaces...179 // 2.4.1 Conditions for Boundedness of Embedding Operators .. 180 // 2.4.2 Criteria for Compactness...182 // 2.4.3 Comments to Sect. 2.4...186 // 2.5 Structure of the Negative Spectrum of the Multidimensional // Schrödinger Operator...188 // 2.5.1 Preliminaries and Notation...188 // 2.5.2 Positivity of the Form 5i [u, u] ...189 // 2.5.3 Semiboundedness of the Schrödinger Operator...190 // 2.5.4 Discreteness of the Negative Spectrum...193 // 2.5.5 Discreteness of the Negative Spectrum of the Operator // Šh for all h...196 // 2.5.6 Finiteness of the Negative Spectrum...197 // 2.5.7 Infiniteness and Finiteness of the Negative Spectrum // of the Operator 5/, for all h...199 // 2.5.8 Proofs of Lemmas 2.5.1/1 and 2.5.1/2...200 // 2.5.9 Comments to Sect. 2.5...203 // 2.6 Properties of Sobolev Spaces Generated by Quadratic Forms // with Variable Coefficients...205 // 2.6.1 Degenerate Quadratic Form ...205 // 2.6.2 Completion in the Metric of a Generalized Dirichlet // Integral...208 // 2.6.3 Comments to Sect. 2.6...212 // 2.7 Dilation Invariant Sharp Hardy’s Inequalities ...213 // 2.7.1 Hardy’s Inequality with Sharp Sobolev Remainder // Term...213 // 2.7.2 Two-Weight Hardy’s Inequalities...214 // 2.7.3 Comments to Sect. 2.7...219 // 2.8 Sharp Hardy-Leray Inequality for Axisymmetric // Divergence-Free Fields...220 // 2.8.1 Statement of Results...220 // 2.8.2 Proof of Theorem 1...222 // 2.8.3 Proof

of Theorem 2...227 // 2.8.4 Comments to Sect. 2.8...229 // Contents xiii // 3 Conductor and Capacitary Inequalities with Applications // to Sobolev-Type Embeddings...231 // 3.1 Introduction...231 // 3.2 Comparison of Inequalities (3.1.4) and (3.1.5)...233 // 3.3 Conductor Inequality (3.1.1)...234 // 3.4 Applications of the Conductor Inequality (3.1.1)...236 // 3.5 p-Capacity Depending on v and Its Applications to a // Conductor Inequality and Inequality (3.4.1)...241 // 3.6 Compactness and Essential Norm...243 // 3.7 Inequality (3.1.10) with Integer I > 2...245 // 3.8 Two-Weight Inequalities Involving Fractional Sobolev Norms.. 249 // 3.9 Comments to Chap. 3...252 // 4 Generalizations for Functions on Manifolds and Topological Spaces...255 // 4.1 Introduction...255 // 4.2 Integral Inequalities for Functions on Riemannian Manifolds .. 257 // 4.3 The First Dirichlet-Laplace Eigenvalue and Isoperimetric // Constant...261 // 4.4 Conductor Inequalities for a Dirichlet-Type Integral with a // Locality Property ...265 // 4.5 Conductor Inequality for a Dirichlet-Type Integral without // Locality Conditions...270 // 4.6 Sharp Capacitary Inequalities and Their Applications...273 // 4.7 Capacitary Improvement of the Faber-Krahn Inequality...278 // 4.8 Two-Weight Sobolev Inequality with Sharp Constant ...282 // 4.9 Comments to Chap. 4...286 // 5 Integrability of Functions in the Space 287 // 5.1 Preliminaries...288 // 5.1.1 Notation...288 // 5.1.2 Lemmas on Approximation of Functions in r(fi)

// and Lp(ß)...289 // 5.2 Classes of Sets j?a, Жа and the Embedding L\\(f2) С Lq(Q) .. 290 // 5.2.1 Classes J0...290 // 5.2.2 Technical Lemma...293 // 5.2.3 Embedding L\\{Í2) С Lq(Q)...295 // 5.2.4 Area Minimizing Function Ам and Embedding of // L\\(Q) into Lq(Q) ...298 // 5.2.5 Example of a Domain in ...299 // 5.3 Subareal Mappings and the Classes ý’а and Жа ...300 // 5.3.1 Subareal Mappings...300 // 5.3.2 Estimate for the Function A in Terms of Subareal // Mappings ...302 // 5.3.3 Estimates for the Function A for Special Domains ...303 // xiv Contents // 5.4 Two-Sided Estimates for the Function Л for the Domain in // Nikodým’s Example ...308 // 5.5 Compactness of the Embedding L\\(Í2) С Lq(fi) (q > 1)...311 // 5.5.1 Class JQ ...311 // 5.5.2 Compactness Criterion...312 // 5.6 Embedding W}r{í2,dí2) C Lq{(2) ...314 // 5.6.1 Class Jťa,0...314 // 5.6.2 Examples of Sets in ...315 // 5.6.3 Continuity of the Embedding Operator // WlT(i2,dQ) Lq{Q) ...316 // 5.7 Comments to Chap. 5...319 // 6 Integrability of Functions in the Space L* (17)...323 // 6.1 Conductivity...324 // 6.1.1 Equivalence of Certain Definitions of Conductivity 324 // 6.1.2 Some Properties of Conductivity...326 // 6.1.3 Dirichlet Principle with Prescribed Level Surfaces and // Its Corollaries...328 // 6.2 Multiplicative Inequality for Functions Which Vanish on a // Subset off?...329 // 6.3 Classes of Sets /Pi0...331 // 6.3.1 Definition and Simple Properties of Ур<а...331 // 6.3.2 Identity of

the Classes and Ja...333 // 6.3.3 Necessary and Sufficient Condition for the Validity of // a Multiplicative Inequality for Functions in W a(Q) ... 334 // 6.3.4 Criterion for the Embedding W’s(fi) с Lq.(f2), // p<q* ...’...336 // 6.3.5 Function t/Д/.Р and the Relationship of the Classes // J p,Q and Ja...337 // 6.3.6 Estimates for the Conductivity Minimizing Function // ťjií.p for Certain Domains...338 // 6.4 Embedding Wp S(í?) c Lq-(Í2) for q* < p...341 // 6.4.1 Estimate for the Norm in Lq-(Q) with q* < p for // Functions which Vanish on a Subset of 1?...341 // 6.4.2 Class J%,a and the Embedding W S(Q) С Lq.(fi) for // 0 < q* < p...342 // 6.4.3 Embedding L*(J?) C Lq-(Í2) for a Domain with Finite // Volume ...343 // 6.4.4 Sufficient Condition for Belonging to Жѵ,а...345 // 6.4.5 Necessary Conditions for Belonging to the Classes // •#p,a and Jťp,a...345 // 6.4.6 Examples of Domains in Жѵ а ...347 // 6.4.7 Other Descriptions of the Classes and Жѵ а 348 // 6.4.8 Integral Inequalities for Domains with Power Cusps... 350 // Contents XV // 6.5 More on the Nikodým Example...352 // 6.6 Some Generalizations...360 // 6.7 Inclusion W r(Q) С Lq{Q) (r > q) for Domains with Infinite // Volume ...364 // 6.7.1 Classes and  p,a...364 // 6.7.2 Embedding IVp r(J?j С Lq(f2) (r > q)...367 // OO // 6.7.3 Example of a Domain in the Class Jp,a...368 // (0) // 6.7.4 Space Lp (J7) and Its Embedding into Lq((2)...370 // 6.7.5 Poincaré-Type Inequality for Domains with Infinite Volume

...371 // 6.8 Compactness of the Embedding Ьр((2) С Lq(Q)...374 // 6.8.1 Class Jp,a...374 // 6.8.2 Compactness Criteria...375 // 6.8.3 Sufficient Conditions for Compactness of the // Embedding Lp(í2) С Lq-((2)...376 // 6.8.4 Compactness Theorem for an Arbitrary Domain with Finite Volume ... 377 // 6.8.5 Examples of Domains in the class Ур<а...378 // 6.9 Embedding Llp{Q) С Lq(í2)...379 // 6.10 Applications to the Neumann Problem for Strongly Elliptic Operators...380 // 6.10.1 Second-Order Operators...381 // 6.10.2 Neumann Problem for Operators of Arbitrary Order ... 382 // 6.10.3 Neumann Problem for a Special Domain...385 // 6.10.4 Counterexample to Inequality (6.10.7) ...389 // 6.11 Inequalities Containing Integrals over the Boundary ...390 // 6.11.1 Embedding Wp r(í2,dí2) C Lq(fž)...390 // 6.11.2 Classes and 393 // 6.11.3 Examples of Domains in ■/’р.«-1 * and Ј?о’~Х)...394 // 6.11.4 Estimates for the Norm in Lq(d(2)...395 // 6.11.5 Class Ур,а_1) and Compactness Theorems...397 // 6.11.6 Criteria of Solvability of Boundary Value Problems for Second-Order Elliptic Equations...399 // 6.12 Comments to Chap. 6...401 // Continuity and Boundedness of Functions in Sobolev // Spaces...405 // 7.1 The Embedding W±{(2) с C(fl) П LX{Q)...406 // 7.1.1 Criteria for Continuity of Embedding Operators of // and Llp{(2) into C(O) П L Q)...406 // 7.1.2 Sufficient Condition in Terms of the Isoperimetric Function for the Embedding W (S2) с C(l?) П L ü) . 409 // xvi Contents

// 7.1.3 Isoperimetric Function and a Brezis-Gallouět- // Wainger-Туре Inequality...410 // 7.2 Multiplicative Estimate for Modulus of a Function in W (Q) . 412 // 7.2.1 Conditions for Validity of a Multiplicative Inequality... 412 // 7.2.2 Multiplicative Inequality in the Limit Case // r — (p - n)/n...414 // 7.3 Continuity Modulus of Functions in Lp(S7)...416 // 7.4 Boundedness of Functions with Derivatives in Birnbaum- // Orlicz Spaces...419 // 7.5 Compactness of the Embedding Wp(i2) С C(fl) П Lx(Í2) ...422 // 7.5.1 Compactness Criterion...422 // 7.5.2 Sufficient Condition for the Compactness in Terms of // the Isoperimetric Function...423 // 7.5.3 Domain for Which the Embedding Operator of Wp{(2) // into C(J?) П Loo(Q) is Bounded but not Compact 424 // 7.6 Generalizations to Sobolev Spaces of an Arbitrary Integer // Order...426 // 7.6.1 The (p,  -Conductivity...426 // 7.6.2 Embedding Llp(í2) С C(Q) П Lx{f2)...427 // 7.6.3 Embedding V (íi) С C(Q) П L fi)...428 // 7.6.4 Compactness of the Embedding // L‘p(f2) C C(í?) n Lx{íi) ...429 // 7.6.5 Sufficient Conditions for the Continuity and the Compactness of the Embedding // Llp{íi) С C(Í2) O Lx(fl) ...430 // 7.6.6 Embedding Operators for the Space Wp(Q) П W£(i2), // I > 2k ...432 // 7.7 Comments to Chap. 7...434 // 8 Localization Moduli of Sobolev Embeddings for General // Domains...435 // 8.1 Localization Moduli and Their Properties ...437 // 8.2 Counterexample for the Case p = q...442 // 8.3 Critical Sobolev

Exponent...444 // 8.4 Generalization ...446 // 8.5 Measures of Noncompactness for Power Cusp-Shaped // Domains ...447 // 8.6 Finiteness of the Negative Spectrum of a Schrödinger // Operator on /?-Cusp Domains ...452 // 8.7 Relations of Measures of Noncompactness with Local // Isoconductivity and Isoperimetric Constants...456 // 8.8 Comments to Chap. 8...457 // Contents xvii // Space of Functions of Bounded Variation ...459 // 9.1 Properties of the Set Perimeter and Functions in BV{Q) 459 // 9.1.1 Definitions of the Space BV(Q) and of the Relative // Perimeter ...459 // 9.1.2 Approximation of Functions in BV(Q)...460 // 9.1.3 Approximation of Sets with Finite Perimeter...463 // 9.1.4 Compactness of the Family of Sets with Uniformly Bounded Relative Perimeters ...464 // 9.1.5 Isoperimetric Inequality...464 // 9.1.6 Integral Formula for the Norm in BV(fi)...465 // 9.1.7 Embedding BV(Q) С Lq(Q)...466 // 9.2 Gauss-Green Formula for Lipschitz Functions...467 // 9.2.1 Normal in the Sense of Federer and Reduced // Boundary ...467 // 9.2.2 Gauss-Green Formula...467 // 9.2.3 Several Auxiliary Assertions...468 // 9.2.4 Study of the Set N...470 // 9.2.5 Relations Between varVx«? and *on9 ...473 // 9.3 Extension of Functions in BV(Q) onto R”...477 // 9.3.1 Proof of Necessity of (9.3.2) ...478 // 9.3.2 Three Lemmas on Pcn(&) ...478 // 9.3.3 Proof of Sufficiency of (9.3.2) ...480 // 9.3.4 Equivalent Statement of Theorem 9.3...482 // 9.3.5 One More Extension Theorem ...483 // 9.4 Exact Constants

for Certain Convex Domains...484 // 9.4.1 Lemmas on Approximations by Polyhedra...484 // 9.4.2 Property of Pen ...486 // 9.4.3 Expression for the Set Function tq(S’) for a Convex Domain...486 // 9.4.4 The Function \\Q\\ for a Convex Domain...487 // 9.5 Rough Trace of Functions in BV(Q) and Certain Integral Inequalities...489 // 9.5.1 Definition of the Rough Trace and Its Properties 489 // 9.5.2 Integrability of the Rough Trace...492 // 9.5.3 Exact Constants in Certain Integral Estimates for the Rough Trace...493 // 9.5.4 More on Integrability of the Rough Trace...495 // 9.5.5 Extension of a Function in BV(f2) to С(2 by // a Constant...496 // 9.5.6 Multiplicative Estimates for the Rough Trace...497 // 9.5.7 Estimate for the Norm in Ln/(n_l-l(Q) of a Function // in BV{Q) with Integrable Rough Trace...499 // 9.6 Traces of Functions in BV(Q) on the Boundary and // Gauss-Green Formula...500 // 9.6.1 Definition of the Trace...500 // xviii Contents // 9.6.2 Coincidence of the Trace and the Rough Trace...501 // 9.6.3 Trace of the Characteristic Function...504 // 9.6.4 Integrability of the Trace of a Function in BV((2)...504 // 9.6.5 Gauss-Green Formula for Functions in BV(ii)...505 // 9.7 Comments to Chap. 9...507 // 10 Certain Function Spaces, Capacities, and Potentials...511 // 10.1 Spaces of Functions Differentiable of Arbitrary Positive Order . 512 // 10.1.1 Spaces w‘p, Wp, blpl Blp for I > 0...512 // 10.1.2 Riesz and Bessel Potential Spaces ...516 // 10.1.3 Other Properties of

the Introduced Function Spaces ... 519 // 10.2 Bourgain, Brezis, and Mironescu Theorem Concerning Limiting Embeddings of Fractional Sobolev Spaces ...521 // 10.2.1 Introduction...521 // 10.2.2 Hardy-Type Inequalities...522 // 10.2.3 Sobolev Embeddings... ...528 // 10.2.4 Asymptotics of the Norm in VVp(R") assjO...528 // 10.3 On the Brezis and Mironescu Conjecture Concerning a Gagliardo-Nirenberg Inequality for Fractional Sobolev Norms . 530 // 10.3.1 Introduction...530 // 10.3.2 Main Theorem...531 // 10.4 Some Facts from Nonlinear Potential Theory...536 // 10.4.1 Capacity cap(e, Slp) and Its Properties...536 // 10.4.2 Nonlinear Potentials...538 // 10.4.3 Metric Properties of Capacity...541 // 10.4.4 Refined Functions...544 // 10.5 Comments to Chap. 10...545 // 11 Capacitary and Trace Inequalities for Functions in Rn // with Derivatives of an Arbitrary Order...549 // 11.1 Description of Results...549 // 11.2 Capacitary Inequality of an Arbitrary Order...552 // 11.2.1 A Proof Based on the Smooth Truncation of // a Potential...552 // 11.2.2 A Proof Based on the Maximum Principle for Nonlinear Potentials...554 // 11.3 Conditions for the Validity of Embedding Theorems in Terms // of Isocapacitary Inequalities ...556 // 11.4 Counterexample to the Capacitary Inequality for the Norm // in ЩП) ...558 // 11.5 Ball and Pointwise Criteria...564 // 11.6 Conditions for Embedding into Lq(p) for p > q > 0...570 // 11.6.1 Criterion in Terms of the Capacity Minimizing // Function...570 // 11.6.2

Two Simple Cases...574 // Contents xix // 11.7 Cartan-Type Theorem and Estimates for Capacities...575 // 11.8 Embedding Theorems for the Space Sp (Conditions in Terms // of Balls, p > 1)...579 // 11.9 Applications...582 // 11.9.1 Compactness Criteria...582 // 11.9.2 Equivalence of Continuity and Compactness of the // Embedding Hlp C Lq(p) for p > q...583 // 11.9.3 Applications to the Theory of Elliptic Operators ... 586 // 11.9.4 Criteria for the Rellich-Kato Inequality...586 // 11.10 Embedding Theorems for p = 1 ...588 // 11.10.1 Integrability with Respect to a Measure...588 // 11.10.2 Criterion for an Upper Estimate of a Difference Seminorm (the Case p = 1)...590 // 11.10.3 Embedding into a Riesz Potential Space...596 // 11.11 Criteria for an Upper Estimate of a Difference Seminorm // (the Case p > 1) ...597 // 11.11.1 Case q> p...597 // 11.11.2 Capacitary Sufficient Condition in the Case q = p ... 603 // 11.12 Comments to Chap. 11...607 // Pointwise Interpolation Inequalities for Derivatives and Potentials ...611 // 12.1 Pointwise Interpolation Inequalities for Riesz and Bessel // Potentials...612 // 12.1.1 Estimate for the Maximal Operator of a Convolution 612 // 12.1.2 Pointwise Interpolation Inequality for Riesz // Potentials...613 // 12.1.3 Estimates for \\ J~wxp\\...614 // 12.1.4 Estimates for jJ-W(S — xp)|...619 // 12.1.5 Pointwise Interpolation Inequality for Bessel // Potentials...620 // 12.1.6 Pointwise Estimates Involving and A1 и 622 // 12.1.7 Application: Weighted

Norm Interpolation // Inequalities for Potentials ...623 // 12.2 Sharp Pointwise Inequalities for Vu ...624 // 12.2.1 The Case of Nonnegative Functions...624 // 12.2.2 Functions with Unrestricted Sign. Main Result 624 // 12.2.3 Proof of Inequality (12.2.6)...626 // 12.2.4 Proof of Sharpness...627 // 12.2.5 Particular Case a>(r) = ra,a >0...634 // 12.2.6 One-Dimensional Case...636 // 12.3 Pointwise Interpolation Inequalities Involving “Fractional // Derivatives”...638 // 12.3.1 Inequalities with Fractional Derivatives on the // Right-Hand Sides ...638 // XX Contents // 12.3.2 Inequality with a Fractional Derivative Operator on // the Left-Hand Side...641 // 12.3.3 Application: Weighted Gagliardo-Nirenberg-Type Inequalities for Derivatives ...643 // 12.4 Application of (12.3.11) to Composition Operator in // Fractional Sobolev Spaces...643 // 12.4.1 Introduction...643 // 12.4.2 Proof of Inequality (12.4.1)...645 // 12.4.3 Continuity of the Map (12.4.2)...648 // 12.5 Comments to Chap. 12...653 // 13 A Variant of Capacity...657 // 13.1 Capacity Cap...657 // 13.1.1 Simple Properties of Cap(e, Llp(fi))...657 // 13.1.2 Capacity of a Continuum...660 // 13.1.3 Capacity of a Bounded Cylinder ...662 // 13.1.4 Sets of Zero Capacity Cap(-, Wp)...663 // 13.2 On (p, /)-Polar Sets...663 // 13.3 Equivalence of Two Capacities...664 // 13.4 Removable Singularities of /-Harmonic Functions in L™ 666 // 13.5 Comments to Chap. 13...668 // 14 Integral Inequality for Functions on a Cube ...669

14.1 Connection Between the Best Constant and Capacity // (Case k = 1)...670 // 14.1.1 Definition of a (p, Z)-Negligible Set...670 // 14.1.2 Main Theorem...670 // 14.1.3 Variant of Theorem 14.1.2 and Its Corollaries 673 // 14.2 Connection Between Best Constant and the (p,/)-Inner // Diameter (Case /с = 1)...675 // 14.2.1 Set Function Apq(G)...675 // 14.2.2 Definition of the (p, Z)-Inner Diameter...676 // 14.2.3 Estimates for the Best Constant in (14.1.3) by the // (p, Z)-Inner Diameter...676 // 14.3 Estimates for the Best Constant С in the General Case 679 // 14.3.1 Necessary and Sufficient Condition for Validity of // the Basic Inequality...679 // 14.3.2 Polynomial Capacities of Function Classes...680 // 14.3.3 Estimates for the Best Constant С in the Basic Inequality...681 // 14.3.4 Class <£o(e) and Capacity Capfc(e, Llp(Q2d))...684 // 14.3.5 Lower Bound for Capfc...685 // 14.3.6 Estimates for the Best Constant in the Case of // Small (p,/)-Inner Diameter...687 // Contents xxi // 14.3.7 A Logarithmic Sobolev Inequality with Application to the Uniqueness Theorem for Analytic Functions in the Class Lp(U) ...689 // 14.4 Comments to Chap. 14...691 // 15 Embedding of the Space Llp(f2) into Other Function // Spaces...693 // 15.1 Preliminaries ...693 // 15.2 Embedding Llp(f2) С 9\\П)...694 // 15.2.1 Auxiliary Assertions...694 // 15.2.2 Case П = Rn...696 // 15.2.3 Case n = pi, p>l ...697 // 15.2.4 Case n < pi and Noninteger n/p...697 // 15.2.5 Case n < pi, 1 < p < oo, and Integer

n/p...698 // 15.3 Embedding Ĺp(í2) C Lq(í2,loс) ...701 // 15.4 Embedding Ĺlp(í2) C Lq(Q) (the Case p < q)...703 // 15.4.1 A Condition in Terms of the (p, /(-Inner Diameter ... 703 // 15.4.2 A Condition in Terms of Capacity...704 // 15.5 Embedding Ĺlp(f2) С Lq{Q) (the Case p > q > 1)...707 // 15.5.1 Definitions and Lemmas...707 // 15.5.2 Basic Result...710 // 15.5.3 Embedding Ĺ‘p(í2) С Lq(f2) for an “Infinite Funnel” . 712 // 15.6 Compactness of the Embedding Ĺlp{Si) C Lq(í2)...714 // 15.6.1 Case p < q ...714 // 15.6.2 Casep> q ...715 // 15.7 Application to the Dirichlet Problem for a Strongly Elliptic // Operator...716 // 15.7.1 Dirichlet Problem with Nonhomogeneous Boundary Data ...717 // 15.7.2 Dirichlet Problem with Homogeneous Boundary // Data ...718 // 15.7.3 Discreteness of the Spectrum of the Dirichlet // Problem ...719 // 15.7.4 Dirichlet Problem for a Nonselfadjoint Operator 719 // 15.8 Applications to the Theory of Quasilinear Elliptic Equations . 721 // 15.8.1 Solvability of the Dirichlet Problem for Quasilinear Equations in Unbounded Domains...721 // 15.8.2 A Weighted Multiplicative Inequality...725 // 15.8.3 Uniqueness of a Solution to the Dirichlet Problem with an Exceptional Set for Equations of Arbitrary Order...727 // 15.8.4 Uniqueness of a Solution to the Neumann Problem // for Quasilinear Second-Order Equation...730 // 15.9 Comments to Chap. 15...733 // xxii Contents // 16 Embedding Ĺlp{íi,u) C W™(f2)...737 // 16.1 Auxiliary Assertions...737

// 16.2 Continuity of the Embedding Operator // Ĺlp(í2,v)-> Wrm(ß) ...739 // 16.3 Compactness of the Embedding Operator // Llp{Q,v) ->W?{Q) ...742 // 16.3.1 Essential Norm of the Embedding Operator ...742 // 16.3.2 Criteria for Compactness...744 // 16.4 Closability of Embedding Operators...746 // 16.5 Application: Positive Definiteness and Discreteness of the Spectrum of a Strongly Elliptic Operator...749 // 16.6 Comments to Chap. 16...751 // 17 Approximation in Weighted Sobolev Spaces...755 // 17.1 Main Results and Applications...755 // 17.2 Capacities...757 // 17.3 Applications of Lemma 17.2/3...761 // 17.4 Proof of Theorem 17.1...765 // 17.5 Comments to Chap. 17...768 // 18 Spectrum of the Schrödinger Operator and the Dirichlet Laplacian...769 // 18.1 Main Results on the Schrödinger Operator ...770 // 18.2 Discreteness of Spectrum: Necessity...773 // 18.3 Discreteness of Spectrum: Sufficiency...781 // 18.4 A Sufficiency Example...783 // 18.5 Positivity of Ну...787 // 18.6 Structure of the Essential Spectrum of Ну...787 // 18.7 Two-Sided Estimates of the First Eigenvalue of the Dirichlet Laplacian ...789 // 18.7.1 Main Result...789 // 18.7.2 Lower Bound...790 // 18.7.3 Upper Bound...795 // 18.7.4 Comments to Chap. 18...800 // References...803 // List of Symbols...849 // Subject Index ...853 // Author Index...859