Theory of Sobolev multipliers : with applications to differential and integral operators

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Theory of Sobolev multipliers : with applications to differential and integral operators / Vladimir G. Maz'ya, Tatyana O. Shaposhnikova

資料種別:

図書

出版情報:

Berlin : Springer, c2009

形態:

xiii, 609 p. ; 25 cm

シリーズ名:

Die Grundlehren der mathematischen Wissenschaften ; v. 337 <BA00007422>

著者名:

ISBN:

9783540694908 [3540694900]

書誌ID:

BA87373736

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書誌詳細

注記:

Includes bibliographical references (p. 591-603) and index

主題:

Differential operators; Integral operators; Multipliers (Mathematical analysis); Sobolev spaces

言語:

英語

目次情報:

Introduction

Description and Properties of Multipliers / Part I：

Trace Inequalities for Functions in Sobolev Spaces / 1：

Trace Inequalities for Functions in <$>w_1^m<$> and <$>W_1^m<$> / 1.1：

The Case m = 1 / 1.1.1：

The Case m ≥ 1 / 1.1.2：

Trace Inequalities for Functions in <$>w_p^m<$> and <$>W_p^m<$>, p > 1 / 1.2：

Preliminaries / 1.2.1：

The (p, m)-Capacity / 1.2.2：

Estimate for the Integral of Capacity of a Set Bounded by a Level Surface / 1.2.3：

Estimates for Constants in Trace Inequalities / 1.2.4：

Other Criteria for the Trace Inequality (1.2.29) with p > 1 / 1.2.5：

The Fefferman and Phong Sufficient Condition / 1.2.6：

Estimate for the L_q-Norm with respect to an Arbitrary Measure / 1.3：

The case 1 ≤ p < q / 1.3.1：

The case q < p ≤ n/m / 1.3.2：

Multipliers in Pairs of Sobolev Spaces / 2：

Characterization of the Space <$>M(W_1^m \to W_1^l)<$> / 2.1：

Characterization of the Space <$>M(W_p^m \to W_p^l)<$> for p > 1 / 2.3：

Another Characterization of the Space <$>M(W_p^m \to W_p^l)<$> for 0 < l < m, pm ≤ n, p > 1 / 2.3.1：

Characterization of the Space <$>M(W_p^m \to W_p^l)<$> for pm > n, p > 1 / 2.3.2：

One-Sided Estimates for Norms of Multipliers in the Case pm ≤ n / 2.3.3：

Examples of Multipliers / 2.3.4：

The Space <$>M(W_p^m\left( {\op R}_+^n \right) \to W_p^l({\op R}_+^n))<$> / 2.4：

Extension from a Half-Space / 2.4.1：

The Case p > 1 / 2.4.2：

The Case p = 1 / 2.4.3：

The Space <$>M(W_p^m \to W_p^{-k})<$> / 2.5：

The Space <$>M(W_p^m \to W_q^l)<$> / 2.6：

Certain Properties of Multipliers / 2.7：

The Space <$>M(w_p^m \to w_p^l)<$> / 2.8：

Multipliers in Spaces of Functions with Bounded Variation / 2.9：

The Spaces Mbv and MBV / 2.9.1：

Multipliers in Pairs of Potential Spaces / 3：

Trace Inequality for Bessel and Riesz Potential Spaces / 3.1：

Properties of Bessel Potential Spaces / 3.1.1：

Properties of the (p, m)-Capacity / 3.1.2：

Main Result / 3.1.3：

Description of <$>M(H_p^m \to H_p^l)<$> / 3.2：

Auxiliary Assertions / 3.2.1：

Imbedding of <$>M(H_p^m \to H_p^l)<$> into <$>M(H_p^{m-l} \to L_p)<$> / 3.2.2：

Estimates for Derivatives of a Multiplier / 3.2.3：

Multiplicative Inequality for the Strichartz Function / 3.2.4：

Auxiliary Properties of the Bessel Kernel G_l / 3.2.5：

Upper Bound for the Norm of a Multiplier / 3.2.6：

Lower Bound for the Norm of a Multiplier / 3.2.7：

Description of the Space <$>M(H_p^m \to H_p^l)<$> / 3.2.8：

Equivalent Norm in <$>M(H_p^m \to H_p^l)<$> Involving the Norm in L_mp/(m-l) / 3.2.9：

Characterization of <$>M(H_p^m \to H_p^l)<$>, m > l, Involving the Norm in L_1,unif / 3.2.10：

The Space <$>M(H_p^m \to H_p^l)<$> for mp > n / 3.2.11：

One-Sided Estimates for the Norm in <$>M(H_p^m \to H_p^l)<$> / 3.3：

Lower Estimate for the Norm in <$>M(H_p^m \to H_p^l)<$> Involving Morrey Type Norms / 3.3.1：

Upper Estimate for the Norm in <$>M(H_p^m \to H_p^l)<$> Involving Marcinkiewicz Type Norms / 3.3.2：

Upper Estimates for the Norm in <$>M(H_p^m \to H_p^l)<$> Involving Norms in <$>H_{n/m}^l<$> / 3.3.3：

Upper Estimates for the Norm in <$>M(H_p^m \to H_p^l)<$> by Norms in Besov Spaces / 3.4：

Properties of the Space <$>B_{q,\infty}^{\mu}<$> / 3.4.1：

Estimates for the Norm in <$>M(H_p^m \to H_p^l)<$> by the Norm in <$>B_{q,\infty}^{\mu}<$> / 3.4.3：

Estimate for the Norm of a Multiplier in <$>MH_p^l({\op R}^1)<$> by the q-Variation / 3.4.4：

Miscellaneous Properties of Multipliers in <$>M(H_p^m \to H_p^l)<$> / 3.5：

Spectrum of Multipliers in <$>H_p^l<$> and <$>H_{p^\prime}^{-l}<$> / 3.6：

Preliminary Information / 3.6.1：

Facts from Nonlinear Potential Theory / 3.6.2：

Main Theorem / 3.6.3：

Proof of Theorem 3.6.1 / 3.6.4：

The Space <$>M(h_p^m \to h_p^l)<$> / 3.7：

Positive Homogeneous Multipliers / 3.8：

The Space <$>M(H_p^m(\partial{\cal B}_1) \to H_p^l(\partial{\cal B}_1))<$> / 3.8.1：

Other Normalizations of the Spaces <$>h_p^m<$> and <$>H_p^m<$> / 3.8.2：

Positive Homogeneous Elements of the Spaces <$>M(h_p^m \to h_p^l)<$> and <$>M(H_p^m \to H_p^l)<$> / 3.8.3：

The Space <$>M(B_p^m \to B_p^l)<$> with p > 1 / 4：

Properties of Besov Spaces / 4.1：

Survey of Known Results / 4.2.1：

Properties of the Operators <$>{\cal D}_{p,l}<$> and D_p,l / 4.2.2：

Pointwise Estimate for Bessel Potentials / 4.2.3：

Proof of Theorem 4.1.1 / 4.3：

Estimate for the Product of First Differences / 4.3.1：

Trace Inequality for <$>B_p^k<$>, p > 1 / 4.3.2：

Auxiliary Assertions Concerning <$>M(B_p^m \to B_p^l)<$> / 4.3.3：

Lower Estimates for the Norm in <$>M(B_p^m \to B_p^l)<$> / 4.3.4：

Proof of Necessity in Theorem 4.1.1 / 4.3.5：

Proof of Sufficiency in Theorem 4.1.1 / 4.3.6：

The Case mp > n / 4.3.7：

Lower and Upper Estimates for the Norm in <$>M(B_p^m \to B_p^l)<$> / 4.3.8：

Sufficient Conditions for Inclusion into <$>M(W_p^m \to W_p^l)<$> with Noninteger m and l / 4.4：

Conditions Involving the Space <$>B_{q,\infty}^{\mu}<$> / 4.4.1：

Conditions Involving the Fourier Transform / 4.4.2：

Conditions Involving the Space <$>B_{q,p}^l<$> / 4.4.3：

Conditions Involving the Space <$>H_{n/m}^l<$> / 4.5：

Composition Operator on <$>M(W_p^m \to W_p^l)<$> / 4.6：

The Space <$>M(B_1^m \to B_1^l)<$> / 5：

Trace Inequality for Functions in <$>B_1^l({\op R}^n)<$> / 5.1：

Auxiliary Facts / 5.1.1：

Properties of Functions in the Space <$>B_1^k({\op R}^n)<$> / 5.1.2：

Trace and Imbedding Properties / 5.2.1：

Auxiliary Estimates for the Poisson Operator / 5.2.2：

Descriptions of <$>M(B_1^m \to B_1^l)<$> with Integer l / 5.3：

A Norm in <$>M(B_1^m \to B_1^l)<$> / 5.3.1：

Description of <$>M(B_1^m \to B_1^l)<$> Involving <$>{\cal D}_{1,l}<$> / 5.3.2：

<$>M(B_1^m({\op R}^n) \to B_1^l({\op R}^n))<$> as the Space of Traces / 5.3.3：

Interpolation Inequality for Multipliers / 5.3.4：

Description of the Space <$>M(B_1^m \to B_1^l)<$> with Noninteger l / 5.4：

Further Results on Multipliers in Besov and Other Function Spaces / 5.5：

Peetre's Imbedding Theorem / 5.5.1：

Related Results on Multipliers in Besov and Triebel-Lizorkin Spaces / 5.5.2：

Multipliers in BMO / 5.5.3：

Maximal Algebras in Spaces of Multipliers / 6：

Pointwise Interpolation Inequalities for Derivatives / 6.1：

Inequalities Involving Derivatives of Integer Order / 6.2.1：

Inequalities Involving Derivatives of Fractional Order / 6.2.2：

Maximal Banach Algebra in <$>M(W_p^m \to W_p^l)<$> / 6.3：

Maximal Banach Algebra in <$>M(W_1^m \to W_1^l)<$> / 6.3.1：

Maximal Algebra in Spaces of Bessel Potentials / 6.4：

Pointwise Inequalities Involving the Strichartz Function / 6.4.1：

Banach Algebra <$>{\cal A}_p^{m,l}<$> / 6.4.2：

Imbeddings ofMaximal Algebras / 6.5：

Essential Norm and Compactness of Multipliers / 7：

Two-Sided Estimates for the Essential Norm. The Case m > l / 7.1：

Estimates Involving Cutoff Functions / 7.2.1：

Estimate Involving Capacity (The Case mp < n, p > 1) / 7.2.2：

Estimates Involving Capacity (The Case mp = n, p > 1) / 7.2.3：

Proof of Theorem 7.0.3 / 7.2.4：

Sharpening of the Lower Bound for the Essential Norm in the Case m > l, mp ≤ n, p > 1 / 7.2.5：

Estimates of the Essential Norm for mp > n, p > 1 and for p = 1 / 7.2.6：

One-Sided Estimates for the Essential Norm / 7.2.7：

The Space of Compact Multipliers / 7.2.8：

Two-Sided Estimates for the Essential Norm in the Case m = l / 7.3：

Estimate for the Maximum Modulus of a Multiplier in <$>W_p^l<$> by its Essential Norm / 7.3.1：

Estimates for the Essential Norm Involving Cutoff Functions (The Case lp ≤ n, p > 1) / 7.3.2：

Estimates for the Essential Norm Involving Capacity (The Case lp ≤ n, p > 1) / 7.3.3：

Two-Sided Estimates for the Essential Norm in the Cases lp > n, p > 1, and p =1 / 7.3.4：

Essential Norm in <$>\ring M W_p^l<$> / 7.3.5：

Traces and Extensions of Multipliers / 8：

Multipliers in Pairs of Weighted Sobolev Spaces in <$>{\op R}_+^n<$> / 8.1：

Characterization of <$>M(W_p^{t,\beta} \to W_p^{s,\alpha})<$> / 8.3：

Auxiliary Estimates for an Extension Operator / 8.4：

Pointwise Estimates for T_γ and ∇T_γ / 8.4.1：

Weighted L_p-Estimates for T_γ and ∇T_γ / 8.4.2：

Trace Theorem for the Space <$>M(W_p^{t,\beta} \to W_p^{s,\alpha})<$> / 8.5：

The Case l < 1 / 8.5.1：

The Case l > 1 / 8.5.2：

Proof of Theorem 8.5.1 for l > 1 / 8.5.3：

Traces of Multipliers on the Smooth Boundary of a Domain / 8.6：

<$>MW_p^l({\op R}^n)<$> as the Space of Traces of Multipliers in the Weighted Sobolev Space <$>W_{p,\beta}^k({\op R}^{n+m})<$> / 8.7：

A Property of Extension Operator / 8.7.1：

Trace and Extension Theorem for Multipliers / 8.7.3：

Extension of Multipliers from <$>{\op R}^n<$> to <$>{\op R}_+^{n+1}<$> / 8.7.4：

Application to the First Boundary Value Problem in a Half-Space / 8.7.5：

Traces of Functions in <$>MW_p^l({\op R}^{n+m})<$> on <$>{\op R}^n / 8.8：

Trace and Extension Theorem / 8.8.1：

Multipliers in the Space of Bessel Potentials as Traces of Multipliers / 8.9：

Bessel Potentials as Traces / 8.9.1：

An Auxiliary Estimate for the Extension Operator <$>{\cal T}<$> / 8.9.2：

<$>MH_p^l<$> as a Space of Traces / 8.9.3：

Sobolev Multipliers in a Domain, Multiplier Mappings and Manifolds / 9：

Multipliers in a Special Lipschitz Domain / 9.1：

Special Lipschitz Domains / 9.1.1：

Description of the Space of Multipliers / 9.1.2：

Extension of Multipliers to the Complement of a Special Lipschitz Domain / 9.2：

Multipliers in a Bounded Domain / 9.3：

Domains with Boundary in the Class C^0,1 / 9.3.1：

Description of Spaces of Multipliers in a Bounded Domain with Boundary in the Class C^0,1 / 9.3.2：

Essential Norm and Compact Multipliers in a Bounded Lipschitz Domain / 9.3.4：

The Space <$>ML_p^1(\Omega)<$> for an Arbitrary Bounded Domain / 9.3.5：

Change of Variables in Norms of Sobolev Spaces / 9.4：

(p, l)-Diffeomorphisms / 9.4.1：

More on (p, l)-Diffeomorphisms / 9.4.2：

A Particular (p, l)-Diffeomorphism / 9.4.3：

(p, l)-Manifolds / 9.4.4：

Mappings <$>T_p^{m,l}<$> of One Sobolev Space into Another / 9.4.5：

Implicit Function Theorems / 9.5：

The Space <$>M(\ring {W}_p^m(\Omega) \to W_p^l(\Omega))<$> / 9.6：

Auxiliary Results / 9.6.1：

Description of the Space <$>M(\ring {W}_p^m(\Omega) \to W_p^l(\Omega))<$> / 9.6.2：

Applications of Multipliers to Differential and Integral Operators / Part II：

Differential Operators in Pairs of Sobolev Spaces / 10：

The Norm of a Differential Operator: <$>W_p^h \to W_p^{h-k}<$> / 10.1：

Coefficients of Operators Mapping <$>W_p^h<$> into <$>W_p^{h-k}<$> as Multipliers / 10.1.1：

A Counterexample / 10.1.2：

Operators with Coefficients Independent of Some Variables / 10.1.3：

Differential Operators on a Domain / 10.1.4：

Essential Norm of a Differential Operator / 10.2：

Fredholm Property of the Schrödinger Operator / 10.3：

Domination of Differential Operators in <$>{\op R}^n<$> / 10.4：

Schrödinger Operator and <$>M (w_2^1 \to w_2^{-1})<$> / 11：

Characterization of <$>M(w_2^1 \to w_2^{-1})<$> and the Schrödinger Operator on <$>w_2^1<$> / 11.1：

A Compactness Criterion / 11.3：

Characterization of <$>M (W_2^1 \to W_2^{-1})<$> / 11.4：

Characterization of the Space <$>M(\ring {w}_2^1 (\Omega) \to w_2^{-1}(\Omega))<$> / 11.5：

Second-Order Differential Operators Acting from <$>w_2^1<$> to <$>w_2^{-1}<$> / 11.6：

Relativistic Schrödinger Operator and <$>M(W_2^{1/2} \to W_2^{-1/2})<$> / 12：

Corollaries of the Form Boundedness Criterion and Related Results / 12.1：

Multipliers as Solutions to Elliptic Equations / 13：

The Dirichlet Problem for the Linear Second-Order Elliptic Equation in the Space of Multipliers / 13.1：

Bounded Solutions of Linear Elliptic Equations as Multipliers / 13.2：

The Case β > 1 / 13.2.1：

The Case β = 1 / 13.2.3：

Solutions as Multipliers from <$>W_{2, w(\rho)}^1 (\Omega)<$> into <$>W_{2,1}^1 (\Omega)<$> / 13.2.4：

Solvability of Quasilinear Elliptic Equations in Spaces of Multipliers / 13.3：

Scalar Equations in Divergence Form / 13.3.1：

Systems in Divergence Form / 13.3.2：

Dirichlet Problem for Quasilinear Equations in Divergence Form / 13.3.3：

Dirichlet Problem for Quasilinear Equations in Nondivergence Form / 13.3.4：

Coercive Estimates for Solutions of Elliptic equations in Spaces of Multipliers / 13.4：

The Case of Operators in <$>{\op R}^n<$> / 13.4.1：

Boundary Value Problem in a Half-Space / 13.4.2：

On the L_∞-Norm in the Coercive Estimate / 13.4.3：

Smoothness of Solutions to Higher Order Elliptic Semilinear Systems / 13.5：

Composition Operator in Classes of Multipliers / 13.5.1：

Improvement of Smoothness of Solutions to Elliptic Semilinear Systems / 13.5.2：

Regularity of the Boundary in L_p-Theory of Elliptic Boundary Value Problems / 14：

Description of Results / 14.1：

Change of Variables in Differential Operators / 14.2：

Fredholm Property of the Elliptic Boundary Value Problem / 14.3：

Boundaries in the Classes <$>M_p^{l-1/p}<$>, <$>W_p^{l-1/p}<$>, and <$>M_p^{l-1/p}(\delta)<$> / 14.3.1：

A Priori L_p-Estimate for Solutions and Other Properties of the Elliptic Boundary Value Problem / 14.3.2：

Some Properties of the Operator <$>{\cal T}<$> / 14.4：

Properties of the Mappings λ and / 14.4.2：

Invariance of the Space <$>W_p^l \cap \ring {W}_p^h<$> Under a Change of Variables / 14.4.3：

The Space <$>W_p^{-k}<$> for a Special Lipschitz Domain / 14.4.4：

Auxiliary Assertions on Differential Operators in Divergence Form / 14.4.5：

Solvability of the Dirichlet Problem in <$>W_p^l(\Omega)<$> / 14.5：

Generalized Formulation of the Dirichlet Problem / 14.5.1：

A Priori Estimate for Solutions of the Generalized Dirichlet Problem / 14.5.2：

Solvability of the Generalized Dirichlet Problem / 14.5.3：

The Dirichlet Problem Formulated in Terms of Traces / 14.5.4：

Necessity ofAssumptions on the Domain / 14.6：

A Domain Whose Boundary is in <$>M_2^{3/2} \cap C^1<$> but does not Belong to <$>M_2^{3/2} (\delta)<$> / 14.6.1：

Necessary Conditions for Solvability of the Dirichlet Problem / 14.6.2：

Boundaries of the Class <$>M_p^{l-1/p} (\delta)<$> / 14.6.3：

Local Characterization of <$>M_p^{l-1/p} (\delta)<$> / 14.7：

Estimates for a Cutoff Function / 14.7.1：

Description of <$>M_p^{l-1/p} (\delta)<$> Involving a Cutoff Function / 14.7.2：

Estimate for s₁ / 14.7.3：

Estimate for s₂ / 14.7.4：

Estimate for s₃ / 14.7.5：

Multipliers in the Classical Layer Potential Theory for Lipschitz Domains / 15：

Solvability of Boundary Value Problems in Weighted Sobolev Spaces / 15.1：

(p, k, α)-Diffeomorphisms / 15.2.1：

Weak Solvability of the Dirichlet Problem / 15.2.2：

Continuity Properties of Boundary Integral Operators / 15.2.3：

Proof of Theorems 15.1.1 and 15.1.2 / 15.4：

Proof of Theorem 15.1.1 / 15.4.1：

Proof of Theorem 15.1.2 / 15.4.2：

Properties of Surfaces in the Class <$>M_p^{\ell}(\delta)<$> / 15.5：

Sharpness of Conditions Imposed on &partial;Ω / 15.6：

Necessity of the Inclusion <$>\partial \Omega \in W_p^{\ell}<$> in Theorem 15.2.1 / 15.6.1：

Sharpness of the Condition <$>\partial \Omega \in B_{\infty,p}^{\ell}<$> / 15.6.2：

Sharpness of the Condition <$>\partial \Omega \in M_p^{\ell} (\delta)<$> in Theorem 15.2.1 / 15.6.3：

Sharpness of the Condition <$>\partial \Omega \in M_p^{\ell}(\delta)<$> in Theorem 15.1.1 / 15.6.4：

Extension to Boundary Integral Equations of Elasticity / 15.7：

Applications of Multipliers to the Theory of Integral Operators / 16：

Convolution Operator in Weighted L₂-Spaces / 16.1：

Calculus of Singular Integral Operators with Symbols in Spaces of Multipliers / 16.2：

Continuity in Sobolev Spaces of Singular Integral Operators with Symbols Depending on x / 16.3：

Function Spaces / 16.3.1：

Description of the Space M(H^m,μ → H^l,μ) / 16.3.2：

Corollaries / 16.3.3：

References

List of Symbols

Author and Subject Index

Introduction

Description and Properties of Multipliers / Part I：

Trace Inequalities for Functions in Sobolev Spaces / 1：

Trace Inequalities for Functions in <$>w_1^m<$> and <$>W_1^m<$> / 1.1：

The Case m = 1 / 1.1.1：

The Case m ≥ 1 / 1.1.2：

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