Abstract
We consider the existence of convolution of Roumieu type ultradistribution with the kernel \(e^{s(1+|x|^2)^{q/2}}, q\ge 1\), \(s\in {\mathbb {R}}\backslash \{0\}\).
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References
Bargetz, C., Ortner, N.: Convolution of vector-valued distributions: a survey and comparison. Diss. Math. 495, 1–51 (2013)
Carmichael, R., Kamiński, A., Pilipović, S.: Boundary Values and Convolution in Ultradistribution Spaces. World Scientific Publishing Co. Pte. Ltd. (2007)
Constantine, G.M., Savits, T.H.: A multivariate Faá di Bruno formula with applications. Trans. Am. Math. Soc. 348, 503–520 (1996)
Debrouwere, A., Vindas, J.: On weighted inductive limits of spaces of ultradifferentiable functions and their duals. Math. Nachr. 292(3), 573–602 (2019)
Dierolf, P., Voigt, J.: Convolution and \({\cal{S}}^{\prime }\)-convolution of distributions. Collect. Math. 29, 185–196 (1978)
Dimovski, P., Pilipović, S., Prangoski, B., Vindas, J.: Convolution of ultradistributions and ultradistribution spaces associated to translation-invariant Banach spaces. Kyoto J. Math. 56, 401–440 (2016)
Dimovski, P., Prangoski, B., Vindas, J.: On a class of translation-invariant spaces of quasianalytic ultradistributions. Novi Sad J. Math. 45, 143–175 (2015)
Gelfand, I.M., Shilov, G.E.: Generalized Functions, Volume 2: Spaces of Fundamental and Generalized Functions. Academic Press, New York (1968)
Kamiński, A., Kovačević, D., Pilipović, S.: The equivalence of various definitions of the convolution of ultradistributions. Trudy Mat. Inst. Steklov. 203, 307–322 (1994)
Komatsu, H.: Ultradistributions, I: structure theorems and a characterization. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 20, 25–105 (1973)
Nicola, F., Rodino, L.: Global Pseudo-Differential Calculus on Euclidean Spaces, vol. 4. Birkhäuser, Basel (2010)
Ortner, N.: On convolvability conditions for distributions. Monatsh. Math. 160, 313–335 (2010)
Ortner, N., Wagner, P.: Distribution-Valued Analytic Functions—Theory and Applications, swk edn. Hamburg (2013)
Pilipović, S., Prangoski, B.: On the convolution of Roumieu ultradistributions through the \(\epsilon \) tensor product. Monatsh. Math. 173, 83–105 (2014)
Pilipović, S., Prangoski, B.: Anti-Wick and Weyl quantization on ultradistribution spaces. J. Math. Pures Appl. 103, 472–503 (2015)
Pilipović, S., Prangoski, B.: Complex powers for a class of infinite order hypoelliptic operators. Diss. Math. 529, 1–58 (2018)
Pilipović, S., Prangoski, B., Vindas, J.: On quasianalytic classes of Gelfand–Shilov type. Parametrix and convolution. J. Math. Pures Appl. 116, 174–210 (2018)
Schaefer, H.H.: Topological Vector Spaces. Springer, New York (1970)
Schwartz, L.: Théorie des distributions á valeurs vectorielles. I. Ann. Inst. Fourier 7, 1–41 (1957)
Shiraishi, R.: On the definition of convolution for distributions. J. Sci. Hiroshima Univ. Ser. A 23, 19–32 (1959)
Wagner, P.: Zur Faltung von Distributionen. Mathematische Annalen 276(3), 467–485 (1987)
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B. Prangoski was partially supported by the bilateral project “Microlocal analysis and applications” funded by the Macedonian and Serbian academies of sciences and arts.
The work presented in this paper is partially supported by Ministry of Education and Science, Republic of Serbia, Project no. 174024.
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Pilipović, S., Prangoski, B. & Vučković, Đ. Convolution with the Kernel \(e^{s\langle x\rangle ^q}, q\ge 1, s>0\) Within Ultradistribution Spaces. Mediterr. J. Math. 18, 164 (2021). https://doi.org/10.1007/s00009-021-01805-6
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DOI: https://doi.org/10.1007/s00009-021-01805-6