Kyoto Journal of Mathematics

Convolution of ultradistributions and ultradistribution spaces associated to translation-invariant Banach spaces

Pavel Dimovski, Stevan Pilipović, Bojan Prangoski, and Jasson Vindas

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We introduce and study a number of new spaces of ultradifferentiable functions and ultradistributions and we apply our results to the study of the convolution of ultradistributions. The spaces of convolutors O'C*(Rd) for tempered ultradistributions are analyzed via the duality with respect to the test function spaces OC*(Rd) introduced in this article. We also study ultradistribution spaces associated to translation-invariant Banach spaces of tempered ultradistributions and use their properties to provide a full characterization of the general convolution of Roumieu ultradistributions via the space of integrable ultradistributions. We show that the convolution of two Roumieu ultradistributions T,SD'{Mp}(Rd) exists if and only if (φSˇ)TD'L1{Mp}(Rd) for every φD{Mp}(Rd).

Article information

Kyoto J. Math. Volume 56, Number 2 (2016), 401-440.

Received: 3 September 2014
Revised: 17 February 2015
Accepted: 8 April 2015
First available in Project Euclid: 10 May 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46F05: Topological linear spaces of test functions, distributions and ultradistributions [See also 46E10, 46E35]
Secondary: 46H25: Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) 46F10: Operations with distributions 46E10: Topological linear spaces of continuous, differentiable or analytic functions

convolution of ultradistributions translation-invariant Banach space of tempered ultradistributions ultratempered convolutors Beurling algebra parametrix method


Dimovski, Pavel; Pilipović, Stevan; Prangoski, Bojan; Vindas, Jasson. Convolution of ultradistributions and ultradistribution spaces associated to translation-invariant Banach spaces. Kyoto J. Math. 56 (2016), no. 2, 401--440. doi:10.1215/21562261-3478916.

Export citation


  • [1] J. J. Betancor, C. Fernández, and A. Galbis, Beurling ultradistributions of $L_{p}$-growth, J. Math. Anal. Appl. 279 (2003), 246–265.
  • [2] A. Beurling, “Sur les intégrales de Fourier absolument convergentes et leur application à une transformation fonctionelle” in IX Congr. Math. Scand., Mercator, Helsinki, 1938, 345–366.
  • [3] R. D. Carmichael, A. Kamiński, and S. Pilipović, Boundary Values and Convolution in Ultradistribution Spaces, Ser. Anal. Appl. Comput. 1, World Sci., Hackensack, N. J., 2007.
  • [4] P. Dimovski, S. Pilipović, and J. Vindas, New distribution spaces associated to translation-invariant Banach spaces, Monatsh. Math. 177 (2015), 495–515.
  • [5] P. Dimovski, S. Pilipović, and J. Vindas, Boundary values of holomorphic functions and heat kernel method in translation-invariant distribution spaces, Complex Var. Elliptic Equ. 60 (2015), 1169–1189.
  • [6] P. Dimovski, B. Prangoski, and D. Velinov, Multipliers and convolutors in the space of tempered ultradistributions, Novi Sad J. Math. 44 (2014), 1–18.
  • [7] J. Horváth, Topological Vector Spaces and Distributions, I, Addison-Wesley, Reading, Mass., 1966.
  • [8] A. Kamiński, D. Kovačević, and S. Pilipović, The equivalence of various definitions of the convolution of ultradistributions, Trudy Mat. Inst. Steklov. 203 (1994), 307–322.
  • [9] A. Kamiński and S. Mincheva-Kamińska, Compatibility conditions and the convolution of functions and generalized functions, J. Funct. Spaces Appl. 2013, no. 356724.
  • [10] J. Kisyński, On Cohen’s proof of the factorization theorem, Ann. Polon. Math. 75 (2000), 177–192.
  • [11] H. Komatsu, Ultradistributions, I: Structure theorems and a characterization, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 20 (1973), 25–105.
  • [12] H. Komatsu, Ultradistributions, II: The kernel theorem and ultradistributions with support in a submanifold, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977), 607–628.
  • [13] H. Komatsu, Ultradistributions, III: Vector-valued ultradistributions and the theory of kernels, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), 653–717.
  • [14] H. Komatsu, “Microlocal analysis in Gevrey classes and in complex domains” in Microlocal Analysis and Applications (Montecatini Terme, 1989), Lecture Notes in Math. 1495, Springer, Berlin, 1991, 161–236.
  • [15] G. Köthe, Topological Vector Spaces, I, Grundlehren Math. Wiss. 159, Springer, New York, 1969.
  • [16] G. Köthe, Topological Vector Spaces, II, Grundlehren Math. Wiss. 237, Springer, New York, 1979.
  • [17] N. Ortner and P. Wagner, Applications of weighted $\mathcal{D}'_{L_{p}}$-spaces to the convolution of distributions, Bull. Polish Acad. Sci. Math. 37 (1989), 579–595.
  • [18] N. Ortner and P. Wagner, On the spaces $\mathcal{O}^{m}_{C}$ of John Horváth, J. Math. Anal. Appl. 415 (2014), 62–74.
  • [19] S. Pilipović, Tempered ultradistributions, Boll. Unione Mat. Ital. 2 (1988), 235–251.
  • [20] S. Pilipović, On the convolution in the space of Beurling ultradistributions, Comment. Math. Univ. St. Paul. 40 (1991), 15–27.
  • [21] S. Pilipović, Characterizations of bounded sets in spaces of ultradistributions, Proc. Amer. Math. Soc. 120 (1994), 1191–1206.
  • [22] S. Pilipović and B. Prangoski, On the convolution of Roumieu ultradistributions through the $\varepsilon$ tensor product, Monatsh. Math. 173 (2014), 83–105.
  • [23] B. Prangoski, Laplace transform in spaces of ultradistributions, Filomat 27 (2013), 747–760.
  • [24] H. H. Schaefer, Topological Vector Spaces, Grad. Texts in Math. 3, Springer, New York, 1970.
  • [25] L. Schwartz, Espaces de fonctions différentiables à valeurs vectorielles, J. Analyse Math. 4 (1954/55), 88–148.
  • [26] L. Schwartz, Théorie des distributions à valeurs vectorielles, I, Ann. Inst. Fourier (Grenoble) 7 (1957), 1–141.
  • [27] L. Schwartz, Théorie des distributions, Hermann, Paris, 1966.
  • [28] R. Shiraishi, On the definition of convolutions for distributions, J. Sci. Hiroshima Univ. Ser. A 23 (1959), 19–32.
  • [29] F. Trèves, Topological Vector Spaces, Distributions and Kernels, Academic Press, New York, 1967.
  • [30] P. Wagner, On convolution in weighted $\mathcal{D}'_{L^{p}}$-spaces, Math. Nachr. 287 (2014), 472–477.