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1.

図書

図書
Vladimir Maz'ya ; [translated by T.O. Shaposhnikova]
出版情報: Berlin : Springer, c2011  xxviii, 866 p. ; 25 cm
シリーズ名: Die Grundlehren der mathematischen Wissenschaften ; v. 342
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2.

図書

図書
Vladimir G. Maz'ya, Tatyana O. Shaposhnikova
出版情報: Berlin : Springer, c2009  xiii, 609 p. ; 25 cm
シリーズ名: Die Grundlehren der mathematischen Wissenschaften ; v. 337
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目次情報: 続きを見る
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 Lq-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 Gl / 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 Lmp/(m-l) / 3.2.9:
Characterization of <$>M(H_p^m \to H_p^l)<$>, m > l, Involving the Norm in L1,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 Dp,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 Tγ and ∇Tγ / 8.4.1:
Weighted Lp-Estimates for Tγ and ∇Tγ / 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<$> as a Space of Traces / 8.9.3:
Sobolev Multipliers in a Domain, Multiplier Mappings and Manifolds / 9:
Multipliers in a Special Lipschitz Domain / 9.1:
Special Lipschitz Domains / 9.1.1:
Description of the Space of Multipliers / 9.1.2:
Extension of Multipliers to the Complement of a Special Lipschitz Domain / 9.2:
Multipliers in a Bounded Domain / 9.3:
Domains with Boundary in the Class C0,1 / 9.3.1:
Description of Spaces of Multipliers in a Bounded Domain with Boundary in the Class C0,1 / 9.3.2:
Essential Norm and Compact Multipliers in a Bounded Lipschitz Domain / 9.3.4:
The Space <$>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<$> / 13.4.1:
Boundary Value Problem in a Half-Space / 13.4.2:
On the L-Norm in the Coercive Estimate / 13.4.3:
Smoothness of Solutions to Higher Order Elliptic Semilinear Systems / 13.5:
Composition Operator in Classes of Multipliers / 13.5.1:
Improvement of Smoothness of Solutions to Elliptic Semilinear Systems / 13.5.2:
Regularity of the Boundary in Lp-Theory of Elliptic Boundary Value Problems / 14:
Description of Results / 14.1:
Change of Variables in Differential Operators / 14.2:
Fredholm Property of the Elliptic Boundary Value Problem / 14.3:
Boundaries in the Classes <$>M_p^{l-1/p}<$>, <$>W_p^{l-1/p}<$>, and <$>M_p^{l-1/p}(\delta)<$> / 14.3.1:
A Priori Lp-Estimate for Solutions and Other Properties of the Elliptic Boundary Value Problem / 14.3.2:
Some Properties of the Operator <$>{\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(Hm,μ → Hl,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:
3.

図書

図書
by M. Š. Birman and M. Z. Solomjak ; [translated by F. A. Cezus ; edited by Lev J. Leifman]
出版情報: Providence, R.I. : American Mathematical Society, 1980  viii, 132p, ; 26cm
シリーズ名: American Mathematical Society translations ; ser. 2, v. 114
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4.

図書

図書
Emmanuel Hebey
出版情報: Berlin : Springer, c1996  x, 115 p. ; 24 cm
シリーズ名: Lecture notes in mathematics ; 1635
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5.

図書

図書
Thomas Runst, Winfried Sickel
出版情報: Berlin ; New York : Walter de Gruyter, 1996  x, 547 p. ; 25 cm
シリーズ名: De Gruyter series in nonlinear analysis and applications ; 3
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6.

図書

図書
Julij A. Dubinskij
出版情報: Dordrecht ; Tokyo : D. Reidel, c1986  161 p. ; 23 cm
シリーズ名: Mathematics and its applications ; East European series
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7.

図書

図書
J.T. Marti ; translation editor, J.R. Whiteman
出版情報: London ; Tokyo : Academic Press, 1986  viii, 211 p. ; 24 cm
シリーズ名: Computational mathematics and applications
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目次情報: 続きを見る
(Chapter Titles): Convolutions and Mollifiers
Sobolev Spaces on One-Dimensional Intervals
Sobolev Spaces Hm(G) on Domains G in Rn
The Inequalities of Poincar, and Friedrichs
Extension Theorems
Imbeddings of Hm(G)
Elliptic Boundary Value Problems
The Finite Element Method for the Solution of Elliptic Boundary Value Problems
References
Subject Index
(Chapter Titles): Convolutions and Mollifiers
Sobolev Spaces on One-Dimensional Intervals
Sobolev Spaces Hm(G) on Domains G in Rn
8.

図書

図書
Alois Kufner
出版情報: Chichester ; New York : Wiley, c1985  115 p. ; 24 cm
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9.

図書

図書
William P. Ziemer
出版情報: New York ; Tokyo : Springer-Verlag, c1989  xvi, 308 p. ; 25 cm
シリーズ名: Graduate texts in mathematics ; 120
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目次情報: 続きを見る
Preliminaries
Sobolev Spaces and Their Basic Properties
Pointwise Behavior of Sobolev Functions
PoincarF Inequalities - A Unified Approach
Functions of Bounded Variation
Bibliography
List of Symbols
Index
Preliminaries
Sobolev Spaces and Their Basic Properties
Pointwise Behavior of Sobolev Functions
10.

図書

図書
Umberto Neri
出版情報: Berlin : Springer-Verlag, 1971  vi, 272 p. ; 26 cm
シリーズ名: Lecture notes in mathematics ; 200
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