Theory of Sobolev Multipliers

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1 Vladimir G. Maz'ya Tatyana O. Shaposhnikova Theory of Sobolev Multipliers With Applications to Differential and Integral Operators ^ Springer

2 Introduction Part I Description and Properties of Multipliers 1 Trace Inequalities for Functions in Sobolev Spaces Trace Inequalities for Functions in w anc * W The Case m = l The Case m > Trace Inequalities for Functions in u; and W, p > Preliminaries The (p, m)-capacity Estimate for the Integral of Capacity of a Set Bounded by a Level Surface Estimates for Constants in Trace Inequalities Other Criteria for the Trace Inequality (1.2.29) with p > The Fefferman and Phong Sufficient Condition Estimate for the Lq-Norm with respect to an Arbitrary Measure The case l<p<q The case q<p< n/m 30 2 Multipliers in Pairs of Sobolev Spaces Introduction Characterization of the Space M(W? -> W{) Characterization of the Space M(W -> W l p) for p > Another Characterization of the Space M(W -> W^) for 0 < I < m, pm <n,p>l Characterization of the Space M{W -> W l p) for pm > n, p> 1 47

3 VI Contents One-Sided Estimates for Norms of Multipliers in the Case pm <n Examples of Multipliers The Space M(W {RD -f Wj(R )) Extension from a Half-Space The Case p > The Case p = The Space M(W m p -> W k p ) The Space M(W -» W l q) Certain Properties of Multipliers The Space Af (iu -> w l p) Multipliers in Spaces of Functions with Bounded Variation The Spaces Mbv and MBV 66 3 Multipliers in Pairs of Potential Spaces Trace Inequality for Bessel and Riesz Potential Spaces Properties of Bessel Potential Spaces Properties of the (p, m)-capacity Main Result Description of M(H -> H l p) Auxiliary Assertions Imbedding of M{H -* H l p) into M(H -< -f L p ) Estimates for Derivatives of a Multiplier Multiplicative Inequality for the Strichartz Function Auxiliary Properties of the Bessel Kernel Gi Upper Bound for the Norm of a Multiplier Lower Bound for the Norm of a Multiplier Description of the Space M(H -> H l p) Equivalent Norm in M(# -> H l p) Involving the Norm in L mp/(m _,) Characterization of M(# -* H' p ), m > I, Involving the Norm in Li, un if The Space MiH -+H l p)iormp>n One-Sided Estimates for the Norm in M(# -> H l p) Lower Estimate for the Norm in M(#^ -> H l p) Involving Morrey Type Norms Upper Estimate for the Norm in MiH -> H l p) Involving Marcinkiewicz Type Norms Upper Estimates for the Norm in M(H > H l p) Involving Norms in H l n. 98 n 3.4 Upper Estimates for the Norm in M{H p -* H l p) by Norms in Besov Spaces Auxiliary Assertions Properties of the Space B% x 103

4 VII Estimates for the Norm in M(# -> H p ) by the Norm in B^ Estimate for the Norm of a Multiplier in MH l p{r l ) by the q-variation Miscellaneous Properties of Multipliers in M(ff > H l p) Ill 3.6 Spectrum of Multipliers in H l l p and H pl Preliminary Information Facts from Nonlinear Potential Theory Main Theorem Proof of Theorem The Space M(/i -> h l p) Positive Homogeneous Multipliers The Space M(ff n p (asi) -> H l p(dbi)) Other Normalizations of the Spaces h and H Positive Homogeneous Elements of the Spaces M{h -+ h l p) and MiH -» H l p) 130 The Space M(B -» B l p) with p > Introduction Properties of Besov Spaces Survey of Known Results Properties of the Operators S) Pi i and D Pi i Pointwise Estimate for Bessel Potentials Proof of Theorem Estimate for the Product of First Differences Trace Inequality for Bp, p > Auxiliary Assertions Concerning M(B > B l p) Lower Estimates for the Norm in M(B -» B l p) Proof of Necessity in Theorem Proof of Sufficiency in Theorem The Case mp>n Lower and Upper Estimates for the Norm in MiB? ^B l p) Sufficient Conditions for Inclusion into M{W -» W l p) with Noninteger m and / Conditions Involving the Space -B ioo Conditions Involving the Fourier Transform Conditions Involving the Space B l qp Conditions Involving the Space H l n, m Composition Operator on M(W -> W p ) 174

5 VIII Contents 5 The Space M(B -> B[) Trace Inequality for Functions in B[(R n ) Auxiliary Facts Main Result Properties of Functions in the Space Bf (K n ) Trace and Imbedding Properties Auxiliary Estimates for the Poisson Operator Descriptions of M(JBJ" -> B[) with Integer / A Norm in M(Bf -> B[) Description of M(B^ -> B[) Involving D h i M(J5J n (M") -> 5((R n )) as the Space of Traces Interpolation Inequality for Multipliers Description of the Space M{B^ -> B[) with Noninteger / Further Results on Multipliers in Besov and Other Function Spaces Peetre's Imbedding Theorem Related Results on Multipliers in Besov and Triebel-Lizorkin Spaces Multipliers in BMO Maximal Algebras in Spaces of Multipliers Introduction Pointwise Interpolation Inequalities for Derivatives Inequalities Involving Derivatives of Integer Order Inequalities Involving Derivatives of Fractional Order Maximal Banach Algebra in M(W -* W l p) The Case p > Maximal Banach Algebra in M{W^ -> W[) Maximal Algebra in Spaces of Bessel Potentials Pointwise Inequalities Involving the Strichartz Function Banach Algebra A ' Imbeddings of Maximal Algebras Essential Norm and Compactness of Multipliers Auxiliary Assertions Two-Sided Estimates for the Essential Norm. The Case m > I Estimates Involving Cutoff Functions Estimate Involving Capacity (The Case mp < n, p > 1) Estimates Involving Capacity (The Case mp = n, p > 1) Proof of Theorem Sharpening of the Lower Bound for the Essential Norm in the Case m > I, mp < n, p > 1 262

6 7.2.6 Estimates of the Essential Norm for mp > n, p > 1 and for p = One-Sided Estimates for the Essential Norm The Space of Compact Multipliers Two-Sided Estimates for the Essential Norm in the Case m = I Estimate for the Maximum Modulus of a Multiplier in W p by its Essential Norm Estimates for the Essential Norm Involving Cutoff Functions (The Case Ip < n, p > 1) Estimates for the Essential Norm Involving Capacity (The Case Ip < n, p > 1) Two-Sided Estimates for the Essential Norm in the Cases Ip > n, p > 1, and p=l Essential Norm in MW l p 281 Traces and Extensions of Multipliers Introduction Multipliers in Pairs of Weighted Sobolev Spaces in R Characterization of M{W*'P -> W < a ) Auxiliary Estimates for an Extension Operator Pointwise Estimates for Tj and VT Weighted L p -Estimates for r 7 and vr Trace Theorem for the Space M{W^0 -* W^a) The Case / < The Case I > Proof of Theorem for I > Traces of Multipliers on the Smooth Boundary of a Domain MW p (R n ) as the Space of Traces of Multipliers in the k Weighted Sobolev Space W p J3 (R n+m ) Preliminaries ' A Property of Extension Operator Trace and Extension Theorem for Multipliers Extension of Multipliers from R n to R" Application to the First Boundary Value Problem in a Half-Space Traces of Functions in MW p (R n+m ) on R n Auxiliary Assertions Trace and Extension Theorem Multipliers in the Space of Bessel Potentials as Traces of Multipliers Bessel Potentials as Traces An Auxiliary Estimate for the Extension Operator T MH' p as a Space of Traces 322 IX

7 Sobolev Multipliers in a Domain, Multiplier Mappings and Manifolds Multipliers in a Special Lipschitz Domain Special Lipschitz Domains Auxiliary Assertions Description of the Space of Multipliers Extension of Multipliers to the Complement of a Special Lipschitz Domain Multipliers in a Bounded Domain Domains with Boundary in the Class C 0 ' Auxiliary Assertions Description of Spaces of Multipliers in a Bounded Domain with Boundary in the Class C o> Essential Norm and Compact Multipliers in a Bounded Lipschitz Domain The Space ML p {Q) for an Arbitrary Bounded Domain Change of Variables in Norms of Sobolev Spaces (p, O-Diffeomorphisms More on (p, 2)-Diffeomorphisms A Particular (p,!)-diffeomorphism (p, I)-Manifolds Mappings T '! of One Sobolev Space into Another Implicit Function Theorems The Space M(Wp(Q) -> W l p{q)) Auxiliary Results Description of the Space M(W" (tf) -> W p (fi)) 369 Part II Applications of Multipliers to Differential and Integral Operators 10 Differential Operators in Pairs of Sobolev Spaces The Norm of a Differential Operator: W -» W^~ k Coefficients of Operators Mapping W p into W p ~ k as Multipliers A Counterexample Operators with Coefficients Independent of Some Variables Differential Operators on a Domain Essential Norm of a Differential Operator Fredholm Property of the Schrodinger Operator Domination of Differential Operators in R n 387

8 11 Schrodinger Operator and M(w^ > w^1) Introduction Characterization of M{w\ > w^1) and the Schrodinger Operator om« A Compactness Criterion Characterization of M{W\ -» Wf 1 ) Characterization of the Space M{wl((2) -> w^1^)) Second-Order Differential Operators Acting from w^ to w^ Relativistic Schrodinger Operator and M(Wj 1/2 -» Wz 1/2 ) Auxiliary Assertions Main Result Corollaries of the Form Boundedness Criterion and Related Results Multipliers as Solutions to Elliptic Equations The Dirichlet Problem for the Linear Second-Order Elliptic Equation in the Space of Multipliers Bounded Solutions of Linear Elliptic Equations as Multipliers Introduction The Case 0 > The Case f3 = Solutions as Multipliers from Wl w{p) {(2) into W^X(G) Solvability of Quasilinear Elliptic Equations in Spaces of Multipliers Scalar Equations in Divergence Form Systems in Divergence Form Dirichlet Problem for Quasilinear Equations in Divergence Form Dirichlet Problem for Quasilinear Equations in Nondivergence Form Coercive Estimates for Solutions of Elliptic equations in Spaces of Multipliers The Case of Operators in E n Boundary Value Problem in a Half-Space On the Loo-Norm in the Coercive Estimate Smoothness of Solutions to Higher Order Elliptic Semilinear Systems Composition Operator in Classes of Multipliers Improvement of Smoothness of Solutions to Elliptic Semilinear Systems 477 XI

9 XII Contents 14 Regularity of the Boundary in L p -Theory of Elliptic Boundary Value Problems Description of Results Change of Variables in Differential Operators Fredholm Property of the Elliptic Boundary Value Problem Boundaries in the Classes MJT 1/p, W p ~ 1/p, and M p - 1/P (6) A Priori L p -Estimate for Solutions and Other Properties of the Elliptic Boundary Value Problem Auxiliary Assertions Some Properties of the Operator T Properties of the Mappings A and x Invariance of the Space W p n W p Under a Change of Variables The Space W~ k for a Special Lipschitz Domain Auxiliary Assertions on Differential Operators in Divergence Form Solvability of the Dirichlet Problem in W l p{q) Generalized Formulation of the Dirichlet Problem A Priori Estimate for Solutions of the Generalized Dirichlet Problem Solvability of the Generalized Dirichlet Problem The Dirichlet Problem Formulated in Terms of Traces Necessity of Assumptions on the Domain A Domain Whose Boundary is in M% /2 D C 1 but does not Belong to M 3/2 2 (<5) Necessary Conditions for Solvability of the Dirichlet Problem Boundaries of the Class M p ~ 1/p (6) Local Characterization of M p " 1/v {5) Estimates for a Cutoff Function Description of M p ~ 1/p (J) Involving a Cutoff Function Estimate for s x Estimate for s Estimate for s Multipliers in the Classical Layer Potential Theory for Lipschitz Domains Introduction Solvability of Boundary Value Problems in Weighted Sobolev Spaces (p, k, a)-diffeomorphisms Weak Solvability of the Dirichlet Problem Main Result 542

10 XIII 15.3 Continuity Properties of Boundary Integral Operators Proof of Theorems and Proof of Theorem Proof of Theorem Properties of Surfaces in the Class M p (5) Sharpness of Conditions Imposed on dq Necessity of the Inclusion dfl W p in Theorem Sharpness of the Condition dq B ( xp Sharpness of the Condition dfl M*'{5) in Theorem Sharpness of the Condition df2 e M e p{5) in Theorem Extension to Boundary Integral Equations of Elasticity Applications of Multipliers to the Theory of Integral Operators Convolution Operator in Weighted I<2-Spaces Calculus of Singular Integral Operators with Symbols in Spaces of Multipliers Continuity in Sobolev Spaces of Singular Integral Operators with Symbols Depending oni Function Spaces Description of the Space M(H m <» -» #' ") Main Result Corollaries 588 References 591 List of Symbols 605 Author and Subject Index 607

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