Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Andersson, K. G.: Propagation of analyticity of solutions of partial differential equations with constant coefficients, Ark. Mat., 8 (1971) 277–302.
Atiyah, M. F., R. Bott and L. Gårding: Lacunas for hyperbolic differential operators with constant coefficients I., Acta Math., 124 (1970) 109–189.
Bony, J. M. et P. Schapira: Problème de Cauchy, existence et prolongement pour les hyperfonctions solutions d'équations hyperboliques non strictes, C. R. Acad. Sci. Paris, 274 (1972) 188–191.
__: Solutions hyperfonctions du problème de Cauchy, Hyperfunctions and Pseudo-differential Equations, Lecture Notes in Mathematics No. 287, Springer, Berlin-Heidelberg-New York, 1973, pp. 82–98.
Chazarain, J.: Opérateurs hyperboliques à caracteristiques de multiplicité constante, to appear.
__: An article which will appear in Sém. Bourbaki (1972/1973).
Courant, R. u. D. Hilbert: Methoden der Mathematischen Physik, II, Springer, Berlin, 1937.
__: Methods of Mathematical Physics, II, Interscience, New York, 1962.
Courant, R. and P. D. Lax: The propagation of discontinuities in wave motion, Proc. Nat. Acad. Sci. U.S.A., 42 (1956) 872–876.
Duistermaat, J. and L. Hörmander: Fourier integral operators II, Acta Math., 128 (1971) 183–269.
Egorov, Yu. V.: Conditions for the solvability of pseudo-differential operators, Dokl. Akad. Nauk USSR, 187 (1969) 1232–1234. (In Russian.)
__: On subelliptic pseudo-differential operators, Dokl. Akad. Nauk USSR, 188 (1969) 20–22. (In Russian.)
Friedrichs, K. O.: Symmetric hyperbolic system of linear differential equations, Comm. Pure Appl. Math., 7 (1954) 345–392.
Friedrichs, K. O. u. H. Lewy: Über die Eindeutigkeit und das Abhängigkeitsgebiet der Lösungen beim Anfangsproblem linearer hyperbolischer Differentialgleichungen, Math. Ann., 98 (1928) 177–195.
Gårding, L.: Linear hyperbolic partial differential equations with constant coefficients, Acta Math., 85 (1950) 1–62.
__: Local hyperbolicity, Israel J. Math., 13 (1972) 65–81.
__: A note which will appear in Israel J. Math. as a supplement to Gårding [2].
Hadamard, J.: Lectures on Cauchy Problem in Linear Partial Differential Equations, Yale Univ. Press, 1923. Reprinted by Dover, New York.
Hörmander, L.: On the theory of general partial differential operators, Acta Math., 94 (1955) 161–184.
__: Linear Partial Differential Operators, Springer, Berlin-Heidelberg-New York, 1963.
__: On the singularities of solutions of partial differential equations, Proc. Int. Conf. Functional Analysis and Related Topics, Univ. of Tokyo Press, Tokyo, 1970, pp.31–40.
Kashiwara, M. and T. Kawai: Micro-hyperbolic pseudo-differential operators I, to appear.
Kawai, T.: Construction of local elementary solutions for linear partial differential operators with real analytic coefficients (I)— The case with real principal symbols—, Publ. RIMS, Kyoto Univ., 7 (1971) 363–397.
__: On the global existence of real analytic solutions of linear differential equations (I), J. Math. Soc. Japan, 24 (1972) 481–517.
Lax, P. D.: Asymptotic solutions of oscillatory initial value problems, Duke Math. J., 24 (1957) 627–646.
Leray, J.: Hyperbolic Differential Equations, The Institute for Advanced Study, Princeton, 1952.
__: Un prolongement de la transformation de Laplace qui transforme la solution unitaire d'un opérateur hyperbolique en sa solution élémentaire, Bull. Soc. Math. Fr., 90 (1962) 39–156.
Leray, J. and Y. Ohya: Èquations et systèmes non-linéaires, hyperboliques non-stricts, Math. Ann., 170 (1967) 167–205.
Ludwig, D.: Exact and asymptotic solutions of the Cauchy problem, Comm. Pure Appl. Math., 13 (1960) 473–508.
Mizohata, S.: Analyticity of solutions of hyperbolic systems with analytic coefficients, Comm. Pure Appl. Math., 14 (1961) 547–559.
__: Some remarks on the Cauchy problem, J. Math. Kyoto Univ., 1 (1961) 109–127.
__: Solutions nulles et solutions non-analytiques, ibid., 1 (1961) 271–302.
Mizohata, S. and Y. Ohya: Sur la condition d'hyperbolicité pour les équations a characteristiques multiples, II, Japanese J. Math., 40 (1971) 63–104.
Nirenberg, L. and F. Treves: On local solvability of linear partial differential equations — Part II. Sufficient conditions, Comm. Pure Appl. Math., 23 (1970) 459–510.
Petrowsky, I. G.: Über das Cauchysche Problem für Systeme von partiellen Differentialgleichungen, Mat. Sbornik, 44 (1937), 815–868.
Riemann, B.: Über die Fortpflanzung ebener Luftwellen von endlicher Schwingusweite, Mathematische Werke, Dover, New York, 1953, pp. 156–175.
Sato, M., T. Kawai and M. Kashiwara: (Refered to as S-K-K[1]) Microfunctions and pseudo-differential equations, Hyperfunctions and Pseudo-differential Equations, Lecture Notes in Mathematics. No. 287, Springer, Berlin-Heidelberg-New York, 1973, pp. 265–529.
__: On the structure of single linear pseudo-differential equations, Proc. Japan Acad., 48 (1972), 643–646.
Treves, F.: Ovcyannikov Theorem and Hyperdifferential Operators, I. M. P. A., Rio-de-Janeiro (Brasil), 1969.
__: Analytic-hypoelliptic partial differential equations of principal type, Comm. Pure Appl. Math., 24 (1971) 537–570.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1975 Springer-Verlag
About this paper
Cite this paper
Kashiwara, M., Kawai, T. (1975). Micro-hyperbolic pseudo-differential operators. In: Pham, F. (eds) Hyperfunctions and Theoretical Physics. Lecture Notes in Mathematics, vol 449. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0062916
Download citation
DOI: https://doi.org/10.1007/BFb0062916
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-07151-8
Online ISBN: 978-3-540-37454-1
eBook Packages: Springer Book Archive