Liouville type theorem for solutions of linear partial differential equations with constant coefficients

Akira Kaneko

Annales Polonici Mathematici (2000)

  • Volume: 74, Issue: 1, page 143-159
  • ISSN: 0066-2216

Abstract

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We discuss existence of global solutions of moderate growth to a linear partial differential equation with constant coefficients whose total symbol P(ξ) has the origin as its only real zero. It is well known that for such equations, global solutions tempered in the sense of Schwartz reduce to polynomials. This is a generalization of the classical Liouville theorem in the theory of functions. In our former work we showed that for infra-exponential growth the corresponding assertion is true if and only if the complex zeros of P(ξ) are absent in a strip at infinity. In this article we discuss the growth in between and present a characterization employing the space of ultradistributions corresponding to the growth.

How to cite

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Kaneko, Akira. "Liouville type theorem for solutions of linear partial differential equations with constant coefficients." Annales Polonici Mathematici 74.1 (2000): 143-159. <http://eudml.org/doc/208363>.

@article{Kaneko2000,
abstract = {We discuss existence of global solutions of moderate growth to a linear partial differential equation with constant coefficients whose total symbol P(ξ) has the origin as its only real zero. It is well known that for such equations, global solutions tempered in the sense of Schwartz reduce to polynomials. This is a generalization of the classical Liouville theorem in the theory of functions. In our former work we showed that for infra-exponential growth the corresponding assertion is true if and only if the complex zeros of P(ξ) are absent in a strip at infinity. In this article we discuss the growth in between and present a characterization employing the space of ultradistributions corresponding to the growth.},
author = {Kaneko, Akira},
journal = {Annales Polonici Mathematici},
keywords = {quasianalytic growth; ultradistribution; infra-exponential growth; Liouville theorem; non-quasianalytic growth},
language = {eng},
number = {1},
pages = {143-159},
title = {Liouville type theorem for solutions of linear partial differential equations with constant coefficients},
url = {http://eudml.org/doc/208363},
volume = {74},
year = {2000},
}

TY - JOUR
AU - Kaneko, Akira
TI - Liouville type theorem for solutions of linear partial differential equations with constant coefficients
JO - Annales Polonici Mathematici
PY - 2000
VL - 74
IS - 1
SP - 143
EP - 159
AB - We discuss existence of global solutions of moderate growth to a linear partial differential equation with constant coefficients whose total symbol P(ξ) has the origin as its only real zero. It is well known that for such equations, global solutions tempered in the sense of Schwartz reduce to polynomials. This is a generalization of the classical Liouville theorem in the theory of functions. In our former work we showed that for infra-exponential growth the corresponding assertion is true if and only if the complex zeros of P(ξ) are absent in a strip at infinity. In this article we discuss the growth in between and present a characterization employing the space of ultradistributions corresponding to the growth.
LA - eng
KW - quasianalytic growth; ultradistribution; infra-exponential growth; Liouville theorem; non-quasianalytic growth
UR - http://eudml.org/doc/208363
ER -

References

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  5. [Kn2] A. Kaneko, Introduction to Hyperfunctions, Univ. of Tokyo Press, 1980-1982 (in Japanese); English translation, Kluwer, 1988. 
  6. [Kn3] A. Kaneko, Liouville type theorem for solutions of infra-exponential growth of linear partial differential equations with constant coefficients, Nat. Sci. Report Ochanomizu Univ. 49 (1998), 1-5. 
  7. [Kw] T. Kawai, On the theory of Fourier hyperfunctions and its applications to partial differential equations with constant coefficients, J. Fac. Sci. Univ. Tokyo Sect. 1A 17 (1970), 467-517. 
  8. [Km] H. Komatsu, Ultradistributions I, ibid. 20 (1973), 25-105. 
  9. [MY] M. Morimoto and K. Yoshino, Some examples of analytic functionals with carrier at the infinity, Proc. Japan Acad. 56 (1980), 357-361. Zbl0471.46024
  10. [P] V. P. Palamodov, From hyperfunctions to analytic functionals, Soviet Math. Dokl. 18 (1977), 975-979. Zbl0377.46033
  11. [PM] Y. S. Park and M. Morimoto, Fourier ultra-hyperfunctions in the Euclidean n-space, J. Fac. Sci. Univ. Tokyo Sect. 1A 20 (1973), 121-127. 
  12. [dR] J. W. de Roever, Hyperfunctional singular support of ultradistributions, ibid. 31 (1985), 585-631. Zbl0606.46027
  13. [SM] P. Sargos et M. Morimoto, Transformation des fonctionnelles analytiques à porteurs non compacts, Tokyo J. Math. 4 (1981), 457-492. Zbl0501.46040
  14. [S] L. Schwartz, Théorie des Distributions, new ed., Hermann, Paris, 1966. Zbl0149.09501
  15. [Z] B. Ziemian, The Mellin transformation and multidimensional generalized Taylor expansions of singular functions, J. Fac. Sci. Univ. Tokyo Sect. 1A 36 (1989), 263-295. Zbl0713.46025

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