Regularization of norms of Lagrangian distributions
Author:
Steven Izen
Journal:
Trans. Amer. Math. Soc. 288 (1985), 363380
MSC:
Primary 58G15
MathSciNet review:
773065
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Abstract: Let be a compact smooth manifold, . Let be a fixed Lagrangian submanifold of . The space of Lagrangian distributions is contained in if . When , just misses . A new inner product is defined on in terms of symbols. This inner product contains " information" in the following sense: Slight regularizations of the Lagrangian distributions are taken, putting them in . The asymptotic behavior of the inner product is examined as the regularizations approach the identity. Three different regularization schemes are presented and, in each case, is found to regulate the growth of the ordinary inner product.
 [Du]
J.
J. Duistermaat, Fourier integral operators, Courant Institute
of Mathematical Sciences, New York University, New York, 1973. Translated
from Dutch notes of a course given at Nijmegen University, February 1970 to
December 1971. MR 0451313
(56 #9600)
 [DGu]
J.
J. Duistermaat and V.
W. Guillemin, The spectrum of positive elliptic operators and
periodic bicharacteristics, Invent. Math. 29 (1975),
no. 1, 39–79. MR 0405514
(53 #9307)
 [DHo]
J.
J. Duistermaat and L.
Hörmander, Fourier integral operators. II, Acta Math.
128 (1972), no. 34, 183–269. MR 0388464
(52 #9300)
 [Gu1]
Victor
Guillemin, Some classical theorems in spectral theory
revisited, Seminar on Singularities of Solutions of Linear Partial
Differential Equations (Inst. Adv. Study, Princeton, N.J., 1977/78) Ann.
of Math. Stud., vol. 91, Princeton Univ. Press, Princeton, N.J., 1979,
pp. 219–259. MR 547021
(81b:58045)
 [Gu2]
, The Leray residue symbol and traces of pseudodifferential operators, M.I.T. Notes, 1981.
 [GuSt]
Victor
Guillemin and Shlomo
Sternberg, Geometric asymptotics, American Mathematical
Society, Providence, R.I., 1977. Mathematical Surveys, No. 14. MR 0516965
(58 #24404)
 [Hoc]
Harry
Hochstadt, Integral equations, John Wiley & Sons, New
YorkLondonSydney, 1973. Pure and Applied Mathematics. MR 0390680
(52 #11503)
 [Horl]
Lars
Hörmander, The spectral function of an elliptic operator,
Acta Math. 121 (1968), 193–218. MR 0609014
(58 #29418)
 [Hor2]
Lars
Hörmander, Fourier integral operators. I, Acta Math.
127 (1971), no. 12, 79–183. MR 0388463
(52 #9299)
 [Iz]
S. Izen, Ph.D. Thesis, M.I.T., 1983.
 [Ni]
Louis
Nirenberg, Lectures on linear partial differential equations,
American Mathematical Society, Providence, R.I., 1973. Expository Lectures
from the CBMS Regional Conference held at the Texas Technological
University, Lubbock, Tex., May 22–26, 1972; Conference Board of the
Mathematical Sciences Regional Conference Series in Mathematics, No. 17. MR 0450755
(56 #9048)
 [Se]
R.
T. Seeley, Complex powers of an elliptic operator, Singular
Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966) Amer. Math.
Soc., Providence, R.I., 1967, pp. 288–307. MR 0237943
(38 #6220)
 [Tr]
François
Trèves, Introduction to pseudodifferential and Fourier
integral operators. Vol. 2, Plenum Press, New YorkLondon, 1980.
Fourier integral operators; The University Series in Mathematics. MR 597145
(82i:58068)
 [Du]
 J. J. Duistermaat, Fourier integral operators, Courant Institute Lecture Notes, New York, 1973. MR 0451313 (56:9600)
 [DGu]
 J. J. Duistermaat and V. W. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math. 29 (1975), 2939. MR 0405514 (53:9307)
 [DHo]
 J. J. Duistermaat and L. Hörmander, Fourier integral operators. II, Acta Math. 128 (1972), 184269. MR 0388464 (52:9300)
 [Gu1]
 V. W. Guillemin, Some classical theorems in spectral theory revisited, Seminar on Singularities, Princeton Univ. Press, Princeton, N.J., 1978, pp. 219259. MR 547021 (81b:58045)
 [Gu2]
 , The Leray residue symbol and traces of pseudodifferential operators, M.I.T. Notes, 1981.
 [GuSt]
 V. W. Guillemin and S. Sternberg, Geometric asymptotics, Math. Surveys, no. 14, Amer. Math. Soc, Providence, R.I., 1977. MR 0516965 (58:24404)
 [Hoc]
 H. Hochstadt, Integral equations, Wiley, New York, 1973. MR 0390680 (52:11503)
 [Horl]
 L. Hörmander, The spectral function of an elliptic operator, Acta Math. 121 (1968), 193218. MR 0609014 (58:29418)
 [Hor2]
 , Fourier integral operators. I, Acta Math. 127 (1971), 79183. MR 0388463 (52:9299)
 [Iz]
 S. Izen, Ph.D. Thesis, M.I.T., 1983.
 [Ni]
 L. Nirenberg, Lectures on linear partial differential equations, CBMS Regional Conf. Ser. Math., no. 17, Amer. Math. Soc., Providence, R.I., 1972. MR 0450755 (56:9048)
 [Se]
 R. T. Seeley, Complex powers of an elliptic operator, Proc. Sympos. Pure Math., Vol. 10, Amer. Math. Soc., Providence, R.I., 1967, pp. 288307. MR 0237943 (38:6220)
 [Tr]
 F. Treves, Introduction to pseudodifferential and Fourier integral operators, Plenum Press, New York, 1980. MR 597145 (82i:58068)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198507730651
PII:
S 00029947(1985)07730651
Article copyright:
© Copyright 1985
American Mathematical Society
