Bulletin of the American Mathematical Society

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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.