Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Analytic wave front sets for solutions of linear differential equations of principal type


Author: Karl Gustav Andersson
Journal: Trans. Amer. Math. Soc. 177 (1973), 1-27
MSC: Primary 35D10; Secondary 35S05
DOI: https://doi.org/10.1090/S0002-9947-1973-0316871-1
MathSciNet review: 0316871
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The propagation of analyticity for solutions u of $ P(x,D)u = f$ is studied, in terms of wave front sets, for a large class of differential operators $ P = P(x,D)$ of principal type. In view of a theorem by L. Hörmander [9], the results obtained imply rather precise results about the surjectivity of the mapping $ P:{C^\infty }(\Omega ) \to {C^\infty }(\Omega )$.


References [Enhancements On Off] (What's this?)

  • [1] K. G. Andersson, Propagation of analyticity of solutions of partial differential equations with constant coefficients, Ark. Mat. 8 (1971), 277-302. MR 0299938 (45:8986)
  • [2] -, Propagation of analyticity for solutions of differential equations of principal type, Bull. Amer. Mtah. Soc. 78 (1972), 479-482. MR 0294840 (45:3908)
  • [3] L. Boutet de Monvel and P. Krée, Pseudo-differential operators and Gevrey classes, Ann. Inst. Fourier (Grenoble) 17 (1967), fasc. 1, 295-323. MR 37 #1760. MR 0226170 (37:1760)
  • [4] A. P. Calderón, Uniqueness in the Cauchy problem for partial differential equations, Amer. J. Math. 80 (1958), 16-36. MR 21 #3675. MR 0104925 (21:3675)
  • [5] J. J. Duistermaat and L. Hörmander, Fourier integral operators. II, Acta Math. 128 (1972), 183-269. MR 0388464 (52:9300)
  • [6] L. Hörmander, Pseudo-differential operator's and hypoelliptic equations, Proc. Sympos. Pure Math., vol. 10, Amer. Math. Soc., Providence, R. I., 1967, pp. 138-183. MR 0383152 (52:4033)
  • [7] -, Fourier integral operators. I, Acta Math. 127 (1971), 79-183. MR 0388463 (52:9299)
  • [8] L. Hörmander, Linear differential operators, Actes Congrès Internat. Math. (Nice, 1970), vol. 1, Gauthier-Villars, Paris, 1971, pp. 121-133. MR 0513000 (58:23766)
  • [9] -, Uniqueness theorems and wave front sets for solutions of linear differential equations with analytic coefficients, Comm. Pure Appl. Math. 24 (1971), 671-703. MR 0294849 (45:3917)
  • [10] -, On the existence and regularity of solutions of linear pseudo-differential equations, Enseignement Math. 17 (1971), 99-163. MR 0331124 (48:9458)
  • [11] T. Kawai, Construction of local elementary solutions for linear partial differential operators with real analytic coefficients. I. The case with real principal symbols, Publ. Res. Inst. Math. Sci. Kyoto 7 (1971), 363-397. MR 0346301 (49:11026)
  • [12] B. Malgrange, Unicité du problème de Cauchy, Séminaire Bourbaki 11ième année: 1958/59, Exposé 178, fasc. 2, Secrétariat mathématique, Paris, 1959. MR 28 #1091. MR 689526 (84m:58023)
  • [13] L. Nirenberg, Pseudo-differential operators, Proc. Sympos. Pure Math., vol. 16, Amer. Math. Soc., Providence, R. I., 1970, pp. 149-167. MR 42 #5108. MR 0270217 (42:5108)
  • [14] M. Sato, Hyperfunctions and partial differential equations, Proc. Internat. Conference on Functional Analysis and Related Topics (Tokyo, 1969), Univ. of Tokyo Press, Tokyo, 1970, pp. 91-94.
  • [15] -, Regularity of hyperfunction solutions of partial differential equations, Actes Congrès Internat. Math. (Nice, 1970), vol. 2, Gauthier-Villars, Paris, 1971, pp. 785-794. MR 0650826 (58:31285)
  • [16] -, On pseudo-differential equations in hyper-function theory, Amer. Math. Soc. Sympos. on Partial Differential Equations, Berkeley, 1971.
  • [17] J. J. Duistermaat, On Carleman estimates for pseudo-differential operators, Invent. Math. 17 (1972), 31-43. MR 0322596 (48:958)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35D10, 35S05

Retrieve articles in all journals with MSC: 35D10, 35S05


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1973-0316871-1
Keywords: Analytic wave front sets, pseudo-differential operators
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society