Theory of Sobolev Multipliers

Transcription

1 Vladimir G. Maz'ya Tatyana 0. Shaposhnikova Theory of Sobolev Multipliers With Applications to Differential and Integral Operators 4) Springer

2 Contents Introduction 1 Part I Description and Properties of Multipliers 1 Trace Inequalities for Functions in Sobolev Spaces Trace Inequalities for Functions in w;'.' and Wr The Oase m = The Oase m > Trace Inequalities for Functions in w and Wr, 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 L q-norm with respect to an Arbitrary Measure The case 1 < p < q The case q < p < n/m 30 2 Multipliers in Pairs of Sobolev Spaces Introduction Characterization of the Space M(Wr Characterization of the Space m (wp wii,) for p > Another Characterization of the Space m(w7,=, for 0 < 1 < m, pm < n, p > Characterization of the Space M (WZ' Wil,) 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 (Wp (18.1_) --> Wp/ (Wii_)) Extension from a Half-Space The Case p > The Case p = The Space M (147,n - Wp-k ) The Space M(Wpm ->W!1) Certain Properties of Multipliers The Space M(wZi wpi ) 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(Hpni HP Auxiliary Assertions Imbedding of M(Hpin - HP into M(Hpnr-1 - Lp) Estimates for Derivatives of a Multiplier Multiplicative Inequality for the Strichartz Function Auxiliary Properties of the Bessel Kernel G Upper Bound for the Norm of a Multiplier Lower Bound for the Norm of a Multiplier Description of the Space M(H' - HP Equivalent Norm in M(Hpm - HP Involving the Norm in Lmpl(m _ i) Characterization of M(Hpin --> HP, m > 1, Involving the Norm in Li,unif The Space M(Hr ---> HP for mp> n One-Sided Estimates for the Norm in M(Hm --> H I ) Lower Estimate for the Norm in M(Hp771 - HP Involving Morrey Type Norms Upper Estimate for the Norm in M(Hprn -+ HP Involving Marcinkiewicz Type Norms Upper Estimates for the Norm in M(Hprn Involving Norms in Heim Upper Estimates for the Norm in M(Hpur ----> HP by Norms in Besov Spaces Auxiliary Assertions Properties of the Space B9 103

4 Contents VII Estimates for the Norm in M(Hpm --> by the Norm in Bq Estimate for the Norm of a Multiplier in MHpi (R1) by the q-variation Miscellaneous Properties of Multipliers in M(Hp --> Spectrum of Multipliers in Hp and H; Preliminary Information Facts from Nonlinear Potential Theory Main Theorem Proof of Theorem The Space M(h7 --> hip) Positive Homogeneous Multipliers The Space.1VI(Hprn (ab i ) 11:,(813 1)) Other Normalizations of the Spaces h p7n and HIT Positive Homogeneous Elements of the Spaces M(hm ) and M(Hm HI ) The Space M(Bm --+ ) with p > Introduction Properties of Besov Spaces Survey of Known Results Properties of the Operators 'p,1 and Dp, Pointwise Estimate for Bessel Potentials Proof of Theorem Estimate for the Product of First Differences Trace Inequality for Bpk, p > Auxiliary Assertions Concerning M(Bpm Lower Estimates for the Norm in M(Bpm ---> Bp1 ) Proof of Necessity in Theorem Proof of Sufficiency in Theorem The Case mp > n Lower and Upper Estimates for the Norm in M(Bpm BpI ) Sufficient Conditions for Inclusion into M(Wpm with Noninteger m and Conditions Involving the Space Bq Conditions Involving the Fourier Transform Conditions Involving the Space Bql Conditions Involving the Space Hni Composition Operator an m(wpm 174

5 VIII Contents 5 The Space M (Bin ---> Trace Inequality for Functions in Bi(R n) Auxiliary Facts Main Result Properties of Functions in the Space Be(R n ) Trace and Imbedding Properties Auxiliary Estimates for the Poisson Operator Descriptions of M(Br BI) with Integer A Norm in M (Bin Description of M (Bin # /31) Involving 01, M(Br(Rn ) --> B i (Rn )) as the Space of Traces Interpolation Inequality for Multipliers Description of the Space M(Br > Bi) 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 B M 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(Wpm The Case p > Maximal Banach Algebra in M(Wim > Maximal Algebra in Spaces of Bessel Potentials Pointwise Inequalities Involving the Strichartz Function Banach Algebra Ar Imbeddings of Maximal Algebras Essential Norm and Compactness of Multipliers Auxiliary Assertions Two-Sided Estimates for the Essential Norm. The Case m > 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 > 1, mp < n, p > 1 262

6 Contents IX 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 = Estimate for the Maximum Modulus of a Multiplier in W i by its Essential Norm P Estimates for the Essential Norm Involving Cutoff Functions (The Case lp < n, p > 1) Estimates for the Essential Norm Involving Capacity (The Case lp < n, p > 1) Two-Sided Estimates for the Essential Norm in the Cases lp > n, p > 1, and p = Essential Norm in.1ü-wil, Traces and Extensions of Multipliers Introduction Multipliers in Pairs of Weighted Sobolev Spaces in RZ Characterization of M(Wp1,13 Wi,,s,a ) Auxiliary Estimates for an Extension Operator Pointwise Estimates for ry and VT-y Weighted Lp-Estimates for T'y and Vry Trace Theorem for the Space M(Wpt,0 --> Wis;a ) The Case 1 < The Case 1> Proof of Theorem for 1> Traces of Multipliers on the Smooth Boundary of a Domain MWp/ (1ltn) as the Space of Traces of Multipliers in the Weighted Sobolev Space Wpk,p(R'+') Preliminaries A Property of Extension Operator Trace and Extension Theorem for Multipliers Extension of Multipliers from 1R' to R ±n Application to the First Boundary Value Problem in a Half-Space Traces of Functions in MWpi (Rn+m ) on Rn 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 MHpi as a Space of Traces 322

7 X Contents 9 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, Auxiliary Assertions Description of Spaces of Multipliers in a Bounded Domain with Boundary in the Class C ' Essential Norm and Compact Multipliers in a Bounded Lipschitz Domain The Space Mg(f2) for an Arbitrary Bounded Domain Change of Variables in Norms of Sobolev Spaces (p, 1)-Diffeomorphisms More an (p, l)-diffeomorphisms A Particular (p, l)-diffeomorphism (p, 1)-Manifolds Mappings 7,1 of One Sobolev Space into Another Implicit Function Theorems The Space mcwpm (2) --- Wil,(f2)) Auxiliary Results Description of the Space me(q) - wp1 (2)) 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: In > wph-k Coefficients of Operators Mapping In into Wph-k as Multipliers A Counterexample Operators with Coefficients Independent of Some Variables Differential Operators an a Domain Essential Norm of a Differential Operator Fredholm Property of the Schrödinger Operator Domination of Differential Operators in R 387

8 Contents XI 11 Schrödinger Operator and M(4 --> w2 1) Introduction Characterization of M(t4 --> w2 1 ) and the Schrödinger Operator an A Compactness Criterion Characterization of M(H/ --> W2 1) Characterization of the Space M(4(2) --> w2 1 (2)) Second-Order Differential Operators Acting from w2 to w Relativistic Schrödinger Operator w-1 / 2 ivj 1 / 2 ) and M 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 ß > The Case ß = Solutions as Multipliers from WZ w(p) (2) into 1472,1 (.2) 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 R 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

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 11,1-11P, Wp1-1/P, and Mji,-1/1) (8) A Priori LP-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 Invariante of the Space W77, fl Te Under a Change of Variables The Space for a Special Lipschitz Domain Auxiliary Assertions on Differential Operators in Divergente Form Solvability of the Dirichlet Problem in I/V,((2) 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 23/2 fl C' but does not Belong to /e 2(ö) Necessary Conditions for Solvability of the Dirichlet Problem Boundaries of the Class 4,1-1 /73 (8) Local Characterization of Mp1-1/73 (8) Estimates for a Cutoff Function Description of Mp1-1/73 (5) Involving a Cutoff Function Estimate for si 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, ce)-diffeomorphisms Weak Solvability of the Dirichlet Problem Main Result 542

10 Contents 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 4(8) Sharpness of Conditions Imposed an 0,f Necessity of the Inclusion 0,2 E Wie, in Theorem Sharpness of the Condition 0(2 E Bf, sp Sharpness of the Condition an E M72,(6) in Theorem Sharpness of the Condition 0(2 E me,(6) in Theorem Extension to Boundary Integral Equations of Elasticity Applications of Multipliers to the Theory of Integral Operators Convolution Operator in Weighted L 2-Spaces Calculus of Singular Integral Operators with Symbols in Spaces of Multipliers Continuity in Sobolev Spaces of Singular Integral Operators with Symbols Depending an x Function Spaces Description of the Space M (H772,4 --> Hfl-`) Main Result Corollaries 588 References 591 List of Symbols 605 Author and Subject Index 607