## Kyoto Journal of Mathematics

### A nonlinear theory of infrahyperfunctions

#### Abstract

We develop a nonlinear theory for infrahyperfunctions (also referred to as quasianalytic (ultra)distributions by L. Hörmander). In the hyperfunction case, our work can be summarized as follows. We construct a differential algebra that contains the space of hyperfunctions as a linear differential subspace and in which the multiplication of real analytic functions coincides with their ordinary product. Moreover, by proving an analogue of Schwartz’s impossibility result for hyperfunctions, we show that this embedding is optimal. Our results fully solve an earlier question raised by M. Oberguggenberger.

#### Article information

Source
Kyoto J. Math., Volume 59, Number 4 (2019), 869-895.

Dates
Accepted: 15 June 2017
First available in Project Euclid: 26 September 2019

https://projecteuclid.org/euclid.kjm/1569484831

Digital Object Identifier
doi:10.1215/21562261-2019-0029

Mathematical Reviews number (MathSciNet)
MR4032202

#### Citation

Debrouwere, Andreas; Vernaeve, Hans; Vindas, Jasson. A nonlinear theory of infrahyperfunctions. Kyoto J. Math. 59 (2019), no. 4, 869--895. doi:10.1215/21562261-2019-0029. https://projecteuclid.org/euclid.kjm/1569484831

#### References

• [1] J. Bonet, R. Meise, and S. N. Melikhov, A comparison of two different ways to define classes of ultradifferentiable functions, Bull. Belg. Math. Soc. Simon Stevin 14 (2007), no. 3, 425–444.
• [2] J.-M. Bony and P. Schapira, “Solutions hyperfonctions du problème de Cauchy” in Hyperfunctions and Pseudo-Differential Equations (Katata, 1971), Lecture Notes in Math. 287, Springer, Berlin, 1973, 82–98.
• [3] R. W. Braun, R. Meise, and B. A. Taylor, Ultradifferentiable functions and Fourier analysis, Results Math. 17 (1990), no. 3–4, 206–237.
• [4] W. Chen and Z. Ditzian, Mixed and directional derivatives, Proc. Amer. Math. Soc. 108 (1990), no. 1, 177–185.
• [5] J.-F. Colombeau, New Generalized Functions and Multiplication of Distributions, North-Holland Math. Stud. 84, North-Holland, Amsterdam, 1984.
• [6] J.-F. Colombeau, Elementary Introduction to New Generalized Functions, North-Holland Math. Stud. 113, North-Holland, Amsterdam, 1985.
• [7] F. Colombini, D. Del Santo, and M. Reissig, On the optimal regularity of coefficients in hyperbolic Cauchy problems, Bull. Sci. Math. 127 (2003), no. 4, 328–347.
• [8] F. Colombini, E. Jannelli, and S. Spagnolo, Well-posedness in the Gevrey classes of the Cauchy problem for a nonstrictly hyperbolic equation with coefficients depending on time, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 10 (1983), no. 2, 291–312.
• [9] F. Colombini and S. Spagnolo, An example of a weakly hyperbolic Cauchy problem not well posed in $C^{\infty }$, Acta Math. 148 (1982), 243–253.
• [10] A. Debrouwere, “Generalized function algebras containing spaces of periodic ultradistributions” in Generalized Functions and Fourier Analysis, Oper. Theory Adv. Appl. 260, Birkhäuser/Springer, Cham, 2017, 59–78.
• [11] A. Debrouwere, H. Vernaeve, and J. Vindas, Optimal embeddings of ultradistributions into differential algebras, Monatsh. Math. 186 (2018), no. 3, 407–438.
• [12] A. Debrouwere and J. Vindas, Solution to the first Cousin problem for vector-valued quasianalytic functions, Ann. Mat. Pura Appl. (4) 196 (2017), no. 6, 1983–2003.
• [13] A. Delcroix, M. F. Hasler, S. Pilipović, and V. Valmorin, Sequence spaces with exponent weights: Realizations of Colombeau type algebras, Dissertationes Math. 447 (2007), 1–56.
• [14] J.-W. de Roever, Hyperfunctional singular support of ultradistributions, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 31 (1985), no. 3, 585–631.
• [15] C. Garetto and M. Ruzhansky, Weakly hyperbolic equations with non-analytic coefficients and lower order terms, Math. Ann. 357 (2013), no. 2, 401–440.
• [16] C. Garetto and M. Ruzhansky, Hyperbolic second order equations with non-regular time dependent coefficients, Arch. Ration. Mech. Anal. 217 (2015), no. 1, 113–154.
• [17] A. Gorny, Contribution à l’étude des fonctions dérivables d’une variable réelle, Acta Math. 71 (1939), 317–358.
• [18] T. Gramchev, Nonlinear maps in spaces of distributions, Math. Z. 209 (1992), no. 1, 101–114.
• [19] L. Hörmander, Between distributions and hyperfunctions, Astérisque 131 (1985), 89–106.
• [20] L. Hörmander, The Analysis of Linear Partial Differential Operators, I: Distribution Theory and Fourier Analysis, 2nd ed., Springer, Berlin, 1990.
• [21] G. Hörmann and M. V. de Hoop, Microlocal analysis and global solutions of some hyperbolic equations with discontinuous coefficients, Acta Appl. Math. 67 (2001), no. 2, 173–224.
• [22] G. Hörmann, M. Oberguggenberger, and S. Pilipović, Microlocal hypoellipticity of linear partial differential operators with generalized functions as coefficients, Trans. Amer. Math. Soc. 358 (2006), no. 8, 3363–3383.
• [23] Y. Ito, Fourier hyperfunctions of general type, J. Math. Kyoto Univ. 28 (1988), no. 2, 213–265.
• [24] K. Junker, Vektorwertige Fourierhyperfunktionen, Ph.D. dissertation, University of Düsseldorf, Düsseldorf, 1978.
• [25] H. Komatsu, On the index of ordinary differential operators, J. Fac. Sci. Univ. Tokyo Sect IA Math. 18 (1971), 379–398.
• [26] H. Komatsu, “Relative cohomology of sheaves of solutions of differential equations” in Hyperfunctions and Pseudo-Differential Equations (Katata, 1971), Lecture Notes in Math. 287, Springer, Berlin, 1973, 192–261.
• [27] H. Komatsu, Ultradistributions, I: Structure theorems and a characterization, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 20 (1973), 25–105.
• [28] H. Komatsu, Ultradistributions, III: Vector-valued ultradistributions and the theory of kernels, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), no. 3, 653–717.
• [29] M. Langenbruch, Ultradifferentiable functions on compact intervals, Math. Nachr. 140 (1989), 109–126.
• [30] A. Martineau, “Les hyperfonctions de M. Sato” in Séminaire Bourbaki, Vol. 6 (1960/1961), no. 214, Soc. Math. France, Paris, 1995, 127–139.
• [31] T. Matsuzawa, A calculus approach to hyperfunctions, I, Nagoya Math. J. 108 (1987), 53–66.
• [32] R. Meise and D. Vogt, Introduction to Functional Analysis, Oxf. Grad. Texts Math. 2, Oxford Univ. Press, New York, 1997.
• [33] M. Morimoto, An Introduction to Sato’s Hyperfunctions, Transl. Math. Monogr. 129, Amer. Math. Soc., Providence, 1993.
• [34] M. Oberguggenberger, Multiplication of Distributions and Applications to Partial Differential Equations, Pitman Res. Notes Math. 259, Longman Scientific and Technical, Wiley, New York, 1992.
• [35] H.-J. Petzsche, Generalized functions and the boundary values of holomorphic functions, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 31 (1984), no. 2, 391–431.
• [36] S. Pilipović, Characterizations of bounded sets in spaces of ultradistributions, Proc. Amer. Math. Soc. 120 (1994), no. 4, 1191–1206.
• [37] S. Pilipović, Generalized hyperfunctions and algebra of megafunctions, Tokyo J. Math. 28 (2005), no. 1, 1–12.
• [38] B. Prangoski, Laplace transform in spaces of ultradistributions, Filomat 27 (2013), no. 5, 747–760.
• [39] M. Sato, Theory of hyperfunctions I, J. Fac. Sci. Univ. Tokyo Sect. I 8 (1959), 139–193. II (1960), 387–437.
• [40] P. Schapira, Théorie des hyperfonctions, Lecture Notes in Math. 126, Springer, Berlin, 1970.
• [41] L. Schwartz, Sur l’impossibilité de la multiplication des distributions, C. R. Math. Acad. Sci. Paris 239 (1954), 847–848.
• [42] T. Takiguchi, On the structure of hyperfunctions and ultradistributions, Publ. Res. Inst. Math. Sci. Kokyuroku 1861 (2013), 71–82.
• [43] F. Trèves, Topological Vector Spaces, Distributions and Kernels, Academic Press, New York, 1967.