On the stability and boundedness properties of operators in the Gelfand–Shilov spaces of Roumieu type

Samia Youcefi; Mohammed Said Souid; Muath Awadalla; Meraa Arab; Samia Youcefi; Mohammed Said Souid; Muath Awadalla; Meraa Arab

doi:10.3934/math.2026223

AIMS Mathematics

2026, Volume 11, Issue 3: 5409-5435. doi: 10.3934/math.2026223

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Research article

On the stability and boundedness properties of operators in the Gelfand–Shilov spaces of Roumieu type

1.
Ecole Superieure en Informatique 08-MAI-1945, Sidi Bel Abbes, Algeria
2.
Department of Economic Sciences, Ibn Khaldoun University of Tiaret, Tiaret 14000, Algeria
3.
Department of Allied Sciences, Faculty of Arts and Science, Al-Ahliyya Amman University, Amman, Jordan
4.
Department of Mathematics and Statistics, College of Science, King Faisal University, Al Ahsa 31982, Saudi Arabia

Received: 05 January 2026 Revised: 17 February 2026 Accepted: 25 February 2026 Published: 03 March 2026
MSC : 46E10, 46F05, 35S05, 46E15

This article presents an extensive study of Gelfand–Shilov spaces of Roumieu type defined via sequences of positive real numbers satisfying natural growth and convexity conditions. We introduce generalized anisotropic spaces $ S_{\{N\}}^{\{M\}}\left(\mathbb{R}^{n}\right) $ and explore their fundamental properties, including stability under differentiation, multiplication, translation, and dilatation operators. We establish boundedness results that demonstrate these spaces form differential subalgebras of the space of continuous functions. These results enrich the functional–analytic framework of ultradifferentiable spaces and provide a foundation for further studies in microlocal analysis and partial differential equations involving ultradistributions.
- function spaces,
- ultradifferentiable functions,
- anisotropic spaces,
- differential operators,
- bounded operators
Citation: Samia Youcefi, Mohammed Said Souid, Muath Awadalla, Meraa Arab. On the stability and boundedness properties of operators in the Gelfand–Shilov spaces of Roumieu type[J]. AIMS Mathematics, 2026, 11(3): 5409-5435. doi: 10.3934/math.2026223

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Abstract

This article presents an extensive study of Gelfand–Shilov spaces of Roumieu type defined via sequences of positive real numbers satisfying natural growth and convexity conditions. We introduce generalized anisotropic spaces $ S_{\{N\}}^{\{M\}}\left(\mathbb{R}^{n}\right) $ and explore their fundamental properties, including stability under differentiation, multiplication, translation, and dilatation operators. We establish boundedness results that demonstrate these spaces form differential subalgebras of the space of continuous functions. These results enrich the functional–analytic framework of ultradifferentiable spaces and provide a foundation for further studies in microlocal analysis and partial differential equations involving ultradistributions.

References

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