Book Review
The AMS does not provide abstracts of book reviews.
You may download the entire review from the links below.
MathSciNet review:
2731653
Full text of review:
PDF
This review is available free of charge.
Book Information:
Authors:
Vladimir G. Maz’ya and
Tatyana O. Shaposhnikova
Title:
Theory of Sobolev multipliers: with applications to differential and integral operators
Additional book information:
Grundlehren der mathematischen Wissenschaften\ (Fundamental Principles of Mathematical Sciences), 337,
Springer-Verlag,
Berlin,
2009,
xiv+614 pp.,
ISBN 978-3-540-69490-8,
$139.00,
hardcover
David R. Adams, On the existence of capacitary strong type estimates in $R^{n}$, Ark. Mat. 14 (1976), no. 1, 125–140. MR 417774, DOI 10.1007/BF02385830
David R. Adams and Lars Inge Hedberg, Function spaces and potential theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 314, Springer-Verlag, Berlin, 1996. MR 1411441, DOI 10.1007/978-3-662-03282-4
S.-Y. A. Chang, J. M. Wilson, and T. H. Wolff, Some weighted norm inequalities concerning the Schrödinger operators, Comment. Math. Helv. 60 (1985), no. 2, 217–246. MR 800004, DOI 10.1007/BF02567411
Björn E. J. Dahlberg, Regularity properties of Riesz potentials, Indiana Univ. Math. J. 28 (1979), no. 2, 257–268. MR 523103, DOI 10.1512/iumj.1979.28.28018
A. Devinatz and I. I. Hirschman Jr., Multiplier transformations on $l^{2,\alpha }$, Ann. of Math. (2) 69 (1959), 575–587. MR 104974, DOI 10.2307/1970023
Charles L. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. (N.S.) 9 (1983), no. 2, 129–206. MR 707957, DOI 10.1090/S0273-0979-1983-15154-6
Kurt Hansson, Imbedding theorems of Sobolev type in potential theory, Math. Scand. 45 (1979), no. 1, 77–102. MR 567435, DOI 10.7146/math.scand.a-11827
I. I. Hirschman Jr., Multiplier transformations. II, Duke Math. J. 28 (1961), 45–56. MR 124693
R. Kerman and E. Sawyer, The trace inequality and eigenvalue estimates for Schrödinger operators, Ann. Inst. Fourier (Grenoble) 36 (1986), no. 4, 207–228 (English, with French summary). MR 867921
V. G. Maz′ja, Classes of domains and imbedding theorems for function spaces, Soviet Math. Dokl. 1 (1960), 882–885. MR 0126152
V. G. Maz′ja, On the theory of the higher-dimensional Schrödinger operator, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 1145–1172 (Russian). MR 0174879
Vladimir G. Maz’ja, Sobolev spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. Translated from the Russian by T. O. Shaposhnikova. MR 817985, DOI 10.1007/978-3-662-09922-3
V. G. Maz′ya and I. E. Verbitsky, Form boundedness of the general second-order differential operator, Comm. Pure Appl. Math. 59 (2006), no. 9, 1286–1329. MR 2237288, DOI 10.1002/cpa.20122
Jaak Peetre, New thoughts on Besov spaces, Duke University Mathematics Series, No. 1, Duke University, Mathematics Department, Durham, N.C., 1976. MR 0461123
John C. Polking, A Leibniz formula for some differentiation operators of fractional order, Indiana Univ. Math. J. 21 (1971/72), 1019–1029. MR 318868, DOI 10.1512/iumj.1972.21.21082
Michael Reed and Barry Simon, Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0493420
Partial differential equations. VII, Encyclopaedia of Mathematical Sciences, vol. 64, Springer-Verlag, Berlin, 1994. Spectral theory of differential operators; A translation of Current problems in mathematics. Fundamental directions. Vol. 64 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989 [ MR1033499 (91h:35229a)]; Translation by T. Zastawniak; Translation edited by M. A. Shubin. MR 1313735, DOI 10.1007/978-3-642-57876-2
Robert S. Strichartz, Multipliers on fractional Sobolev spaces, J. Math. Mech. 16 (1967), 1031–1060. MR 0215084
I. E. Verbitsky, Nonlinear potentials and trace inequalities, The Maz′ya anniversary collection, Vol. 2 (Rostock, 1998) Oper. Theory Adv. Appl., vol. 110, Birkhäuser, Basel, 1999, pp. 323–343. MR 1747901
References
- D. R. Adams, On the existence of capacitary strong type estimates in $R^ {n}$, Ark. Mat. 14 (1976), 125–140. MR 0417774 (54:5822)
- D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Springer-Verlag, Berlin, 1996. MR 1411441 (97j:46024)
- S. Y. A. Chang, J. M. Wilson, and T.H. Wolff, Some weighted norm inequalities concerning the Schrödinger operators, Comment. Math. Helv. 60 (1985), 217–246. MR 800004 (87d:42027)
- B. E. J. Dahlberg, Regularity properties of Riesz potentials, Indiana Univ. Math. J., 28 (1979), 257–268. MR 523103 (80g:31004)
- A. Devinatz and I. I. Hirshman, Multiplier transformations on $l^{2, \alpha }$, Ann. Math. 69 (1959), 575–587. MR 0104974 (21:3722)
- C. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. 9 (1983), 129–206. MR 707957 (85f:35001)
- K. Hansson, Imbedding theorems of Sobolev type in potential theory, Math. Scand. 45 (1979), 77–102. MR 567435 (81j:31007)
- I. I. Hirshman, On multiplier transformations, II, Duke Math. J. 28 (1961), 45–56. MR 0124693 (23:A2004)
- R. Kerman and E. Sawyer, The trace inequality and eigenvalue estimates for Schrödinger operators, Ann. Inst. Fourier, Grenoble 36 (1987), 207–228. MR 867921 (88b:35150)
- V. G. Maz’ya, Classes of domains and embedding theorems for functional spaces, Dokl. Akad. Nauk SSSR, 133 (1960), 527–530. MR 0126152 (23:A3448)
- V. G. Maz’ya, On the theory of the $n$-dimensional Schrödinger operator, Izv. Akad. Nauk SSSR, Ser. Matem., 28 (1964), 1145–1172. MR 0174879 (30:5070)
- V. G. Maz’ya, Sobolev Spaces, Springer-Verlag, Berlin–Heidelberg–New York, 1985 (new edition in press). MR 817985 (87g:46056)
- V. G. Maz’ya and I. E. Verbitsky, Form boundedness of the general second order differential operator, Comm. Pure Appl. Math. 59 (2006), 1286–1329. MR 2237288 (2008d:47089)
- J. Peetre, New Thoughts on Besov Spaces, Duke Univ. Press, Durham, NC, 1976. MR 0461123 (57:1108)
- J. C. Polking, A Leibniz formula for some differential operators of fractional order, Indiana Univ. Math. J. 27 (1972), 1019–1029. MR 0318868 (47:7414)
- M. Reed and B. Simon, Methods of Modern Mathematical Physics. II: Fourier Analysis, Self-Adjointness, Academic Press, New York–London, 1975. MR 0493420 (58:12429b)
- G. V. Rozenblum, M. A. Shubin, and M. Z. Solomyak, Spectral Theory of Differential Operators, Encyclopaedia of Math. Sci., 64. Partial Differential Equations VII. (M. A. Shubin, editor). Springer-Verlag, Berlin–Heidelberg, 1994. MR 1313735 (95j:35156)
- R. S. Strichartz, Multipliers of fractional Sobolev spaces, J. Math. Mech. 16 (1967), 1031–1060. MR 0215084 (35:5927)
- I. E. Verbitsky, Nonlinear potentials and trace inequalities, The Maz’ya Anniversary Collection, Vol. 2. Operator Theory Adv. Appl. 110, Birkhäuser, Basel (1999), 323–343. MR 1747901 (2001g:46086)
Review Information:
Reviewer:
I. E. Verbitsky
Affiliation:
Columbia, Missouri
Email:
verbitskyi@missouri.edu
Journal:
Bull. Amer. Math. Soc.
48 (2011), 101-105
DOI:
https://doi.org/10.1090/S0273-0979-10-01297-8
Published electronically:
March 10, 2010
Additional Notes:
The author was partially supported by NSF Grant DMS-0901550.
Review copyright:
© Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
