Book Review
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MathSciNet review:
1567295
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Book Information:
Authors:
Alexander Nagel and
E. M. Stein
Title:
Lectures on pseudo-differential operators: Regularity Theorems and applications to non-elliptic problems
Additional book information:
Mathematical Notes, Princeton Univ. Press, Princeton, N. J., 1979, 159 pp., $6.75.
S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623–727. MR 125307, DOI 10.1002/cpa.3160120405
M. F. Atiyah and I. M. Singer, The index of elliptic operators on compact manifolds, Bull. Amer. Math. Soc. 69 (1963), 422–433. MR 157392, DOI 10.1090/S0002-9904-1963-10957-X
Richard Beals, A general calculus of pseudodifferential operators, Duke Math. J. 42 (1975), 1–42. MR 367730
Richard Beals, $L^{p}$ and Hölder estimates for pseudodifferential operators: sufficient conditions, Ann. Inst. Fourier (Grenoble) 29 (1979), no. 3, vii, 239–260 (English, with French summary). MR 552967
Richard Beals and Charles Fefferman, Spatially inhomogeneous pseudodifferential operators. I, Comm. Pure Appl. Math. 27 (1974), 1–24. MR 352747, DOI 10.1002/cpa.3160270102
Louis Boutet de Monvel, Opérateurs pseudo-différentiels analytiques et problèmes aux limites elliptiques, Ann. Inst. Fourier (Grenoble) 19 (1969), no. 2, 169–268 (1970). MR 437965
A.-P. Calderón, Uniqueness in the Cauchy problem for partial differential equations, Amer. J. Math. 80 (1958), 16–36. MR 104925, DOI 10.2307/2372819
A.-P. Calderón, Singular integrals, Bull. Amer. Math. Soc. 72 (1966), 427–465. MR 209918, DOI 10.1090/S0002-9904-1966-11492-1
A. P. Calderon and A. Zygmund, On the existence of certain singular integrals, Acta Math. 88 (1952), 85–139. MR 52553, DOI 10.1007/BF02392130
A.-P. Calderón and A. Zygmund, Singular integral operators and differential equations, Amer. J. Math. 79 (1957), 901–921. MR 100768, DOI 10.2307/2372441
Ronald R. Coifman and Guido Weiss, Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Mathematics, Vol. 242, Springer-Verlag, Berlin-New York, 1971 (French). Étude de certaines intégrales singulières. MR 0499948
H. O. Cordes and E. A. Herman, Gel′fand theory of pseudo differential operators, Amer. J. Math. 90 (1968), 681–717. MR 454743, DOI 10.2307/2373478
Eugene B. Fabes, Singular integrals and partial differential equations of parabolic type, Studia Math. 28 (1966/67), 81–131. MR 213744, DOI 10.4064/sm-28-1-81-131
G. B. Folland and E. M. Stein, Estimates for the $\bar \partial _{b}$ complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 27 (1974), 429–522. MR 367477, DOI 10.1002/cpa.3160270403
Lars Hörmander, Hypoelliptic differential operators, Ann. Inst. Fourier (Grenoble) 11 (1961), 477–492, XVI (English, with French summary). MR 126071
Lars Hörmander, Pseudo-differential operators, Comm. Pure Appl. Math. 18 (1965), 501–517. MR 180740, DOI 10.1002/cpa.3160180307
Lars Hörmander, Pseudo-differential operators and hypoelliptic equations, Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966) Amer. Math. Soc., Providence, R.I., 1967, pp. 138–183. MR 0383152
L. Hörmander, The Weyl calculus of pseudodifferential operators, Comm. Pure Appl. Math. 32 (1979), no. 3, 360–444. MR 517939, DOI 10.1002/cpa.3160320304
Roger Howe, Quantum mechanics and partial differential equations, J. Functional Analysis 38 (1980), no. 2, 188–254. MR 587908, DOI 10.1016/0022-1236(80)90064-6
B. Frank Jones Jr., A class of singular integrals, Amer. J. Math. 86 (1964), 441–462. MR 161099, DOI 10.2307/2373175
21. A. Knapp and E. M. Stein, Singular integrals and the principal series, Proc. Nat. Acad. Sci. U.S.A. 63 (1969), 281-284.
J. J. Kohn and L. Nirenberg, An algebra of pseudo-differential operators, Comm. Pure Appl. Math. 18 (1965), 269–305. MR 176362, DOI 10.1002/cpa.3160180121
Adam Korányi and István Vági, Intégrales singulières sur certains espaces homogènes, C. R. Acad. Sci. Paris Sér. A-B 268 (1969), A765–A768 (French). MR 286944
Alexander Nagel and E. M. Stein, A new class of pseudo-differential operators, Proc. Nat. Acad. Sci. U.S.A. 75 (1978), no. 2, 582–585. MR 487603, DOI 10.1073/pnas.75.2.582
L. Nirenberg, Pseudo-differential operators, Global Analysis (Proc. Sympos. Pure Math., Vol. XVI, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 149–167. MR 0270217
26. L. Nirenberg and F. Treves, On local solvability of linear partial differential equations. I, Comm. Pure Appl. Math. 23 (1970), 1-38; II, ibid., 459-509; correction, ibid. 24 (1971), 279-288.
R. T. Seeley, Singular integrals on compact manifolds, Amer. J. Math. 81 (1959), 658–690. MR 110022, DOI 10.2307/2372921
R. T. Seeley, Refinement of the functional calculus of Calderón and Zygmund, Nederl. Akad. Wetensch. Proc. Ser. A 68=Indag. Math. 27 (1965), 521–531. MR 0226450
R. T. Seeley, Elliptic singular integral equations, Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966) Amer. Math. Soc., Providence, R.I., 1967, pp. 308–315. MR 0234107
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
M. A. Šubin, Psevdodifferentsial′nye operatory i spektral′naya teoriya, “Nauka”, Moscow, 1978 (Russian). MR 509034
M. A. Šubin, Almost periodic pseudodifferential operators, and von Neumann algebras, Trudy Moskov. Mat. Obšč. 35 (1976), 103–164 (Russian). MR 0636409
Michael Taylor, Pseudo differential operators, Lecture Notes in Mathematics, Vol. 416, Springer-Verlag, Berlin-New York, 1974. MR 0442523
André Unterberger, Symboles associés aux champs de repères de la forme symplectique, C. R. Acad. Sci. Paris Sér. A-B 285 (1977), no. 16, A1005–A1008 (French, with English summary). MR 517103
André Unterberger and Juliane Bokobza, Les opérateurs de Calderon-Zygmund précisés, C. R. Acad. Sci. Paris 259 (1964), 1612–1614 (French). MR 176360
36. G. Weiss and S. Wainger, eds., Harmonic analysis in Euclidean spaces, Proc. Sympos. Pure Math., vol. 35, Amer. Math. Soc., Providence, R. I., 1978.
- 1.
- S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623-727. MR 0125307
- 2.
- M. Atiyah and I. Singer, The index of elliptic operators on compact manifolds, Bull. Amer. Math. Soc. 69 (1963), 422-433. MR 157392
- 3.
- R. Beals, A general calculus of pseudodifferential operators, Duke Math. J. 42 (1975), 1-42. MR 367730
- 4.
- R. Beals, L. II. Sufficient conditions, Ann. Inst. Fourier (Grenoble) 29 (1979), 239-260. MR 552967
- 5.
- R. Beals and Fefferman, Spatially inhomogeneous pseudodifferential operators, Comm. Pure Appl. Math. 27 (1974), 1-24. MR 352747
- 6.
- L. Boutet de Monvel, Opérateurs pseudo-différentiels analytiques et problèmes aux limites elliptiques, Ann. Inst. Fourier (Grenoble) 19 (1969), 163-268. MR 437965
- 7.
- A. P. Calderón, Uniqueness of the Cauchy problem for partial differential equations, Amer. J. Math. 80 (1958), 16-36. MR 104925
- 8.
- A. P. Calderón, Singular integrals, Bull. Amer. Math. Soc. 72 (1966), 427-465. MR 209918
- 9.
- A. P. Calderón and A. Zygmund, On the existence of certain singular integrals, Acta Math. 88 (1952), 85-139. MR 52553
- 10.
- A. P. Calderón and and A. Zygmund, Singular integral operators and differential equations, Amer. J. Math. 79 (1957), 901-921. MR 100768
- 11.
- R. Coifman and G. Weiss, Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Math., no 242, Springer-Verlag, Berlin, 1971. MR 499948
- 12.
- H. O. Cordes and E. Herman, Gelfand theory of pseudo-differential operators, Amer. J. Math. 90 (1968), 681-717. MR 454743
- 13.
- E. Fabes and N. Rivière, Singular integrals with mixed homogeneity, Studia Math. 27 (1966), 19-38. MR 213744
- 14.
- G. Folland and E. M. Stein, Estimates for the $\bar \partial \sbb$ complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 27 (1974), 429-522. MR 367477
- 15.
- L. Hörmander, Hypoelliptic differential operators, Ann. Inst Fourier (Grenoble) 11 (1961), 477-492. MR 126071
- 16.
- L. Hörmander, Pseudo-differential operators, Comm. Pure Appl. Math. 18 (1965), 501-517. MR 180740
- 17.
- L. Hörmander, Pseudo-differential operators and hypoelliptic equations, Proc. Sympos. Pure Math., vol. 10, Amer. Math. Soc., Providence, R. I., 1966, pp. 138-183. MR 383152
- 18.
- L. Hörmander, The Weyl calculus of pseudodifferential operators, Comm. Pure Appl. Math. 32 (1979), 359-443. MR 517939
- 19.
- R. Howe, Quantum mechanics and partial differential equations (preprint). MR 587908
- 20.
- F. Jones, A class of singular integrals, Amer. J. Math. 86 (1964), 441-462. MR 161099
- 21.
- A. Knapp and E. M. Stein, Singular integrals and the principal series, Proc. Nat. Acad. Sci. U.S.A. 63 (1969), 281-284.
- 22.
- J. J. Kohn and L. Nirenberg, An algebra of pseudo-differential operators, Comm. Pure Appl. Math. 18 (1965), 269-305. MR 176362
- 23.
- A. Korányi and S. Vági, Integrales singulières sur certains espaces homogènes, C. R. Acad. Sci. Paris Sér. A-B 268 (1969), 765-768. MR 286944
- 24.
- A. Nagel and E. M. Stein, A new class of pseudodifferential operators, Proc. Nat. Acad. Sci. U.S.A. 75 (1978), 582-585. MR 487603
- 25.
- L. Nirenberg, Pseudo-differential operators, Proc. Sympos. Pure Math., vol. 16, Amer. Math. Soc., Providence, R. I., 1970, pp. 147-168. MR 270217
- 26.
- L. Nirenberg and F. Treves, On local solvability of linear partial differential equations. I, Comm. Pure Appl. Math. 23 (1970), 1-38; II, ibid., 459-509; correction, ibid. 24 (1971), 279-288.
- 27.
- R. Seeley, Singular integrals on compact manifolds, Amer. J. Math. 81 (1959), 658-690. MR 110022
- 28.
- R. Seeley, Refinement of the functional calculus of Calderón and Zygmund, Proc. Akad. Wet. Ned. Ser. A 68 (1965), 521-531. MR 226450
- 29.
- R. Seeley, Elliptic singular integral equations, Proc. Sympos. Pure Math., vol. 10, Amer. Math. Soc., Providence, R. I., 1967, pp. 308-315. MR 234107
- 30.
- E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, N. J., 1970. MR 290095
- 31.
- M. A. Šubin, Pseudodifferential operators and spectral theory "Nauka", Moscow, 1978. (Russian) MR 509034
- 32.
- M. A. Šubin, Pseudodifferential almost-periodic operators and von Neumann algebras, Trans. Moscow Math. Soc. 1979, no. 1 (35), 103-166. MR 636409
- 33.
- M. Taylor, Pseudo-differential operators, Lecture Notes in Math., no. 416, Springer-Verlag, Berlin, 1974. MR 442523
- 34.
- A. Unterberger, Symboles associés aux champs de repères de la forme symplectique, C. R. Acad. Sci. Paris Sér. A-B 285 (1977), A1005-A1008. MR 517103
- 35.
- A. Unterberger and J. Bokobza, Les opérateurs de Caldéron-Zygmund précisés, C. R. Acad. Sci. Paris Sér. A-B 260 (1965), A3265-A3267. MR 176361
- 36.
- G. Weiss and S. Wainger, eds., Harmonic analysis in Euclidean spaces, Proc. Sympos. Pure Math., vol. 35, Amer. Math. Soc., Providence, R. I., 1978.
Review Information:
Reviewer:
Richard Beals
Journal:
Bull. Amer. Math. Soc.
3 (1980), 1069-1074
DOI:
https://doi.org/10.1090/S0273-0979-1980-14859-4