Monatshefte für Mathematik

, Volume 186, Issue 3, pp 407–438 | Cite as

Optimal embeddings of ultradistributions into differential algebras

  • Andreas Debrouwere
  • Hans Vernaeve
  • Jasson Vindas


We construct embeddings of spaces of non-quasianalytic ultradistributions into differential algebras enjoying optimal properties in view of a Schwartz type impossibility result, also shown in this article. We develop microlocal analysis in these algebras consistent with the microlocal analysis in the corresponding spaces of ultradistributions.


Generalized functions Colombeau algebras Multiplication of ultradistributions Wave front sets Ultradifferentiable functions Denjoy–Carleman classes 

Mathematics Subject Classification

Primary 46F05 46F30 Secondary 35A18 


  1. 1.
    Aragona, J., Juriaans, S.O., Colombeau, J.-F.: Locally convex topological algebras of generalized functions: compactness and nuclearity in a nonlinear context. Trans. Am. Math. Soc. 367, 5399–5414 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bengel, G., Schapira, P.: Décomposition microlocale analytique des distributions. Ann. Inst. Fourier (Grenoble) 29, 101–124 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Benmeriem, K., Bouzar, C.: Generalized Gevrey ultradistributions. N. Y. J. Math. 15, 37–72 (2009)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Benmeriem, K., Bouzar, C.: An algebra of generalized Roumieu ultradistributions. Rend. Sem. Mat. Univ. Politec. Torino 70, 101–109 (2012)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Beurling, A.: Sur les intégrales de Fourier absolument convergentes et leur application à une transformation fonctionelle. In: IX Congr. Math. Scand., pp. 345–366. Helsingfors (1938)Google Scholar
  6. 6.
    Björck, G.: Linear partial differential operators and generalized distributions. Ark. Mat. 6, 351–407 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Braun, R.W.: An extension of Komatsu’s second structure theorem for ultradistributions. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 40, 411–417 (1993)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Braun, R.W., Meise, R., Taylor, B.A.: Ultradifferentiable functions and Fourier analysis. Results Math. 17, 206–237 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Carmichael, R.D., Kamiński, A., Pilipović, S.: Boundary Values and Convolution in Ultradistribution Spaces. Series on Analysis, Applications and Computation, 1. World Scientific Publishing Co. Pte. Ltd., Hackensack (2007)zbMATHGoogle Scholar
  10. 10.
    Chen, W., Ditzian, Z.: Mixed and directional derivatives. Proc. Am. Math. Soc. 108, 177–185 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cho, J., Kim, K.W.: Real version of Paley–Wiener–Schwartz theorem for ultradistributions with ultradifferentiable singular support. Bull. Korean Math. Soc. 36, 483–493 (1999)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Ciorǎnescu, I., Zsidó, L.: \(\omega \)-ultradistributions and their applications to operator theory. In: Żelazko, W. (ed.) Spectral Theory (Warsaw, 1977), pp. 77–220. Banach Center Publ. 8, PWN, Warsaw (1982).Google Scholar
  13. 13.
    Colombeau, J.-F.: New Generalized Functions and Multiplication of Distributions. North-Holland Publishing Co., Amsterdam (1984)zbMATHGoogle Scholar
  14. 14.
    Colombeau, J.-F.: Elementary Introduction to New Generalized Functions. North-Holland Publishing Co, Amsterdam (1985)zbMATHGoogle Scholar
  15. 15.
    Dapić, N., Pilipović, S., Scarpalézos, D.: Microlocal analysis of Colombeau’s generalized functions: propagation of singularities. J. Anal. Math. 75, 51–66 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Debrouwere, A., Vindas, J.: Discrete characterizations of wave front sets of Fourier–Lebesgue and quasianalytic type. J. Math. Anal. Appl. 438, 889–908 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Delcroix, A., Hasler, M.F., Pilipović, S., Valmorin, V.: Embeddings of ultradistributions and periodic hyperfunctions in Colombeau type algebras through sequence spaces. Math. Proc. Camb. Philos. Soc. 137, 697–708 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Delcroix, A., Hasler, M.F., Pilipović, S., Valmorin, V.: Sequence spaces with exponent weights. Realizations of Colombeau type algebras. Diss. Math. 447, 56 (2007)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Fernández, C., Galbis, A., Jornet, D.: Pseudodifferential operators of Beurling type and the wave front set. J. Math. Anal. Appl. 340, 1153–1170 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Garetto, C.: On hyperbolic equations and systems with non-regular time dependent coefficients. J. Differ. Equ. 259, 5846–5874 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Garetto, C., Ruzhansky, M.: Hyperbolic second order equations with non-regular time dependent coefficients. Arch. Ration. Mech. Anal. 217, 113–154 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Garetto, C., Vernaeve, H.: Hilbert \(\tilde{{\mathbb{C}}}\)-modules: structural properties and applications to variational problems. Trans. Am. Math. Soc. 363, 2047–2090 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Gramchev, T.: Nonlinear maps in spaces of distributions. Math. Z. 209, 101–114 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Grosser, M., Kunzinger, M., Oberguggenberger, M., Steinbauer, R.: Geometric Theory of Generalized Functions. Kluwer Academic Publishers, Dordrecht (2001)zbMATHGoogle Scholar
  25. 25.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis, 2nd edn. Grundlehren der Mathematischen Wissenschaften 256. Springer, Berlin (1990)Google Scholar
  26. 26.
    Hörmann, G.: Hölder–Zygmund regularity in algebras of generalized functions. Z. Anal. Anwend. 23, 139–165 (2004)CrossRefzbMATHGoogle Scholar
  27. 27.
    Hörmann, G., de Hoop, M.V.: Microlocal analysis and global solutions of some hyperbolic equations with discontinuous coefficients. Acta Appl. Math. 67, 173–224 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Hörmann, G., Oberguggenberger, M., Pilipović, S.: Microlocal hypoellipticity of linear partial differential operators with generalized functions as coefficients. Trans. Am. Math. Soc. 358, 3363–3383 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Kolmogorov, A.N.: On inequalities between the upper bounds of successive derivatives of an arbitrary function on an infinite interval (translation). In: American Mathematical Society Translations, Series 1. Number Theory and Analysis, vol. 2, pp. 233–243. American Mathematical Society, Providence (1962)Google Scholar
  30. 30.
    Komatsu, H.: Ultradistributions I. Structure theorems and a characterization. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 20, 25–105 (1973)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Komatsu, H.: Relative cohomology of sheaves of solutions of differential equations. In: Komatsu, H. (ed.) Hyperfunctions and Pseudo-Differential Equations. Lecture Notes in Mathematics, vol. 287, pp. 192–261. Springer, Berlin (1973)CrossRefGoogle Scholar
  32. 32.
    Komatsu, H.: Ultradistributions. III. Vector-valued ultradistributions and the theory of kernels. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29, 653–717 (1982)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Komatsu, H.: Microlocal analysis in Gevrey classes and in complex domains. In: Cattabriga, L., Rodino, L. (eds.) Microlocal Analysis and Applications. Lecture Notes in Mathematics, vol. 1495, pp. 161–236. Springer, Berlin (1991)CrossRefGoogle Scholar
  34. 34.
    Kunzinger, M., Steinbauer, R.: Generalized pseudo-Riemannian geometry. Trans. Am. Math. Soc. 354, 4179–4199 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Nedeljkov, M., Pilipović, S., Scarpalézos, D.: The Linear Theory of Colombeau Generalized Functions, Pitman Research Notes in Mathematics Series, vol. 385. Longman, Harlow (1998)zbMATHGoogle Scholar
  36. 36.
    Oberguggenberger, M.: Multiplication of Distributions and Applications to Partial Differential Equations. Pitman Research Notes in Mathematics, vol. 259. Longman Scientific and Technical, New York (1992)zbMATHGoogle Scholar
  37. 37.
    Oberguggenberger, M., Pilipović, S., Scarpalézos, D.: Local properties of Colombeau generalized functions. Math. Nachr. 256, 88–99 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Pilipović, S.: Characterization of bounded sets in spaces of ultradistributions. Proc. Am. Math. Soc. 120, 1191–1206 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Pilipović, S.: Microlocal analysis of ultradistributions. Proc. Am. Math. Soc. 126, 105–113 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Pilipović, S., Scarpalézos, D.: Colombeau generalized ultradistributions. Math. Proc. Camb. Philos. Soc. 130, 541–553 (2001)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Pilipović, S., Scarpalezos, D.: Regularity properties of distributions and ultradistributions. Proc. Am. Math. Soc. 129, 3531–3537 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Pilipović, S., Scarpalézos, D., Vindas, J.: Regularity properties of distributions through sequences of functions. Monatshefte Math. 170, 227–237 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Roumieu, C.: Sur quelques extensions de la notion de distribution. Ann. Sci. École Norm. Sup. Sér 3(77), 41–121 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Schwartz, L.: Sur l’impossibilité de la multiplication des distributions. C. R. Acad. Sci. Paris 239, 847–848 (1954)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Takiguchi, T.: Structure of quasi-analytic ultradistributions. Publ. Res. Inst. Math. Sci. 43, 425–442 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Wien 2017

Authors and Affiliations

  1. 1.Department of MathematicsGhent UniversityGhentBelgium

Personalised recommendations