Theory of Sobolev Multipliers: With Applications to Differential and Integral Operators (Grundlehren der mathematischen Wissenschaften (337))

The purpose of this book is to give a comprehensive exposition of the theory of pointwise multipliers acting in pairs of spaces of differentiable functions. The theory was essentially developed by the authors during the last thirty years and the present volume is mainly based on their results. Part I is devoted to the theory of multipliers and encloses the following topics: trace inequalities, analytic characterization of multipliers, relations between spaces of Sobolev multipliers and other function spaces, maximal subalgebras of multiplier spaces, traces and extensions of multipliers, essential norm and compactness of multipliers, and miscellaneous properties of multipliers. Part II concerns several applications of this theory: continuity and compactness of differential operators in pairs of Sobolev spaces, multipliers as solutions to linear and quasilinear elliptic equations, higher regularity in the single and double layer potential theory for Lipschitz domains, regularity of the boundary in $L_p$-theory of elliptic boundary value problems, and singular integral operators in Sobolev spaces. Contents......Page 5 Introduction......Page 14 Part I: Description and Properties of Multipliers......Page 17 1.1 Trace Inequalities for Functions in w[sub(m)][sup(1)] and W[sup(m)][sub(1)]......Page 18 1.2 Trace Inequalities for Functions in w[sup(m)][sub(p)] and W[sup(m)][sub(p)] , p > 1......Page 25 1.3 Estimate for the L[sub(q)]-Norm with respect to an Arbitrary Measure......Page 40 2.1 Introduction......Page 43 2.2 Characterization of the Space M(W[sup(m)][sub(1)] → W[sup(l)][sub(1)])......Page 45 2.3 Characterization of the Space M(W[sup(m)[sub(p)] → W[sup(l)][sub(p)]) for p > 1......Page 48 2.4 The Space M(W[sup(m)][sub(p)] (R[sup(n)][sub(+)]) → Wlp(R[sup(n)][su(+)])......Page 60 2.5 The Space M(W[sup(m)][sub(p)] → W–[sup(k)][sub(p)])......Page 64 2.6 The Space M(W[sup(m)][sub(p)] → W[sup(l)][sub(q)])......Page 67 2.7 Certain Properties of Multipliers......Page 68 2.8 The Space M(w[sup(m)][sub(p)] → w[sup(l)][sub(p)])......Page 70 2.9 Multipliers in Spaces of Functions with Bounded Variation......Page 73 3.1 Trace Inequality for Bessel and Riesz Potential Spaces......Page 79 3.2 Description of M(Hmp → H[sup(l)][sub(p)])......Page 85 3.3 One-Sided Estimates for the Norm in M(H[sup(m)][sub(p)] → H[sup(l)][sub(p)])......Page 105 3.4 Upper Estimates for the Norm in M(H[sup(m)][sub(p)] → H[sup(l)][sub(p)]) by Norms in Besov Spaces......Page 109 3.5 Miscellaneous Properties of Multipliers in M(H[sup(m)][sub(p)] → H[sup(l)][sub(p)])......Page 121 3.6 Spectrum of Multipliers in H[sup(l)][sub(p)] and H[sup(−l)][sub(p)]......Page 125 3.7 The Space M(h[sup(m)][sub(p)] → h[sup(l)][sub(p)])......Page 132 3.8 Positive Homogeneous Multipliers......Page 135 4.1 Introduction......Page 143 4.2 Properties of Besov Spaces......Page 144 4.3 Proof of Theorem 4.1.1......Page 151 4.4 Sufficient Conditions for Inclusion into M(W[sup(m)][sub(p)] → W[sup(l)][sub(p)]) with Noninteger m and l......Page 176 4.5 Conditions Involving the Space H[sup(l)][sub(n/m)]......Page 183 4.6 Composition Operator on M(W[sup(m)][sub(p)] → W[sup(l)][sub(p)])......Page 184 5.1 Trace Inequality for Functions in B[sup(l)][sub(1)](R[sup(n)])......Page 189 5.2 Properties of Functions in the Space B[sup(k)][sub(1)] (R[sup(n)])......Page 195 5.3 Descriptions of M(B[sup(m)][sub(1)] → B[sup(l)][sub(1)]) with Integer l......Page 203 5.4 Description of the Space M(B[sup(m)][sub(1)] → B[sup(l)][sup(1)]) with Noninteger l......Page 213 5.5 Further Results on Multipliers in Besov and Other Function Spaces......Page 216 6.1 Introduction......Page 223 6.2 Pointwise Interpolation Inequalities for Derivatives......Page 224 6.3 Maximal Banach Algebra in M(W[sup(m)][sub(p)] → W[sup(l)][sub(p)])......Page 230 6.4 Maximal Algebra in Spaces of Bessel Potentials......Page 237 6.5 Imbeddings of Maximal Algebras......Page 243 7. Essential Norm and Compactness of Multipliers......Page 250 7.1 Auxiliary Assertions......Page 252 7.2 Two-Sided Estimates for the Essential Norm. The Case m > l......Page 257 7.3 Two-Sided Estimates for the Essential Norm in the Case m = l......Page 279 8.2 Multipliers in Pairs of Weighted Sobolev Spaces in R[sup(n)][sub(+)]......Page 294 8.3 Characterization of M(W[sup(t,β)][sub(p)] → W[sup(s,α)][sub(p)])......Page 297 8.4 Auxiliary Estimates for an Extension Operator......Page 301 8.5 Trace Theorem for the Space M(W[sup(t,β)][sub(p)] → W[sup(s,α)][sub(p)])......Page 306 8.6 Traces of Multipliers on the Smooth Boundary of a Domain......Page 313 8.7 MW[sup(l)][sub(p)](R[sup(n)]) as the Space of Traces of Multipliers in the Weighted Sobolev Space W[sup(k)][sub(p,β)] (R[sup(n+m)])......Page 314 8.8 Traces of Functions in MW[sup(l)][sub(p)](R[sup(n+m)]) on R[sup(n)]......Page 321 8.9 Multipliers in the Space of Bessel Potentials as Traces of Multipliers......Page 328 9. Sobolev Multipliers in a Domain, Multiplier Mappings and Manifolds......Page 334 9.1 Multipliers in a Special Lipschitz Domain......Page 335 9.2 Extension of Multipliers to the Complement of a Special Lipschitz Domain......Page 341 9.3 Multipliers in a Bounded Domain......Page 345 9.4 Change of Variables in Norms of Sobolev Spaces......Page 359 9.5 Implicit Function Theorems......Page 373 9.6 The Space M(W[sup(m)][sub(p)](Ω) → W[sup(l)][sub(p)](Ω))......Page 376 Part II: Applications of Multipliers to Differential and Integral Operators......Page 380 10.1 The Norm of a Differential Operator: W[sup(h)][sub(p)] → W[sup(h−k)][sub(p)]......Page 381 10.2 Essential Norm of a Differential Operator......Page 392 10.3 Fredholm Property of the Schrödinger Operator......Page 394 10.4 Domination of Differential Operators in R[sup(n)]......Page 395 11.1 Introduction......Page 398 11.2 Characterization of M(w[sup(1)][sub(2)] → w[sup(−1)][sub(2)]) and the Schrödinger Operator on w[sup(1)][sub(2)]......Page 400 11.3 A Compactness Criterion......Page 414 11.4 Characterization of M(W[sup(1)][sub(2)] → W[sup(−1)][sub(2)])......Page 418 11.5 Characterization of the Space M(W[sup(1)][sub(2)](Ω) → w[sup(−1)][sub(2)] (Ω))......Page 423 12.1 Auxiliary Assertions......Page 434 12.2 Corollaries of the Form Boundedness Criterion and Related Results......Page 448 13.1 The Dirichlet Problem for the Linear Second-Order Elliptic Equation in the Space of Multipliers......Page 452 13.2 Bounded Solutions of Linear Elliptic Equations as Multipliers......Page 454 13.3 Solvability of Quasilinear Elliptic Equations in Spaces of Multipliers......Page 463 13.4 Coercive Estimates for Solutions of Elliptic equations in Spaces of Multipliers......Page 474 13.5 Smoothness of Solutions to Higher Order Elliptic Semilinear Systems......Page 481 14.1 Description of Results......Page 486 14.2 Change of Variables in Differential Operators......Page 488 14.3 Fredholm Property of the Elliptic Boundary Value Problem......Page 490 14.4 Auxiliary Assertions......Page 496 14.5 Solvability of the Dirichlet Problem in W[sup(l)][sub(p)](Ω)......Page 509 14.6 Necessity of Assumptions on the Domain......Page 514 14.7 Local Characterization of M[sup(l−1/p)][sub(p)] (δ)......Page 520 15.1 Introduction......Page 538 15.2 Solvability of Boundary Value Problems in Weighted Sobolev Spaces......Page 544 15.3 Continuity Properties of Boundary Integral Operators......Page 554 15.4 Proof of Theorems 15.1.1 and 15.1.2......Page 558 15.5 Properties of Surfaces in the Class M[sup(l)][sub(p)](δ)......Page 566 15.6 Sharpness of Conditions Imposed on ∂Ω......Page 569 15.7 Extension to Boundary Integral Equations of Elasticity......Page 575 16.1 Convolution Operator in Weighted L[sup(2)]-Spaces......Page 579 16.2 Calculus of Singular Integral Operators with Symbols in Spaces of Multipliers......Page 581 16.3 Continuity in Sobolev Spaces of Singular Integral Operators with Symbols Depending on x......Page 585 References......Page 596 List of Symbols......Page 609 E......Page 611 M......Page 612 Z......Page 613 'I never heard of "Ugli?cation," Alice ventured to say. 'What is it?'' Lewis Carroll, "Alice in Wonderland" Subject and motivation. The present book is devoted to a theory of m- tipliers in spaces of di?erentiable functions and its applications to analysis, partial di?erential and integral equations. By a multiplier acting from one functionspaceS intoanotherS, wemeanafunctionwhichde?nesabounded 1 2 linear mapping ofS intoS by pointwise multiplication. Thus with any pair 1 2 of spacesS, S we associate a third one, the space of multipliersM(S?S ) 1 2 1 2 endowed with the norm of the operator of multiplication. In what follows, the role of the spacesS andS is played by Sobolev spaces, Bessel potential 1 2 spaces, Besov spaces, and the like. The Fourier multipliers are not dealt with in this book. In order to emp- size the di?erence between them and the multipliers under consideration, we attach Sobolev's name to the latter. By coining the term Sobolev multipliers we just hint at various spaces of di?erentiable functions of Sobolev's type, being fully aware that Sobolev never worked on multipliers. After all, Fourier never did either. "The purpose of this book is to give a comprehensive exposition of the theory of pointwise multipliers acting in pairs of spaces of differentiable functions. The theory was essentially developed by the authors during the last thirty years and the present volume is mainly based on their results."--BOOK JACKET Vladimir G. Maz'ya, Tatyana O. Shaposhnikova. Includes Bibliographical References (p. 591-603) And Index.