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Let A(Ω) denotes the classical space of real analytic functions defined on a non empty open set Ω of the Euclidean space Rn. Different classes of ultradifferentiable functions have been considered and studied as an extension or a generalisation of the space A(Ω). The following scheme illustrates some ones of this development :
↗ E ω (Ω)
A (Ω) → E M (Ω)
↘E s,Γ (Ω)
The ultradifferentiable function classes are denoted as follows
1. E M (Ω) is the Denjoy-Carleman space, see [14],[15],[20].
2. E ω (Ω) is the Beurling space, see [1], [?], [13].
3. E s,Γ (Ω) is the multi-anisotropic Gevrey space, see [16], [22].
Remark 1 It is not necessary to remind the importance of these spaces in mathematics, and particularly in partial differential equations, [19], [17] and [21].
A recent paper [2] addressed the issue to find the relationship between the spaces EM (Ω) and Eω(Ω) . So, firstly this work deals with the question of finding a relationship between the Denjoy-Carleman space E M (Ω) and the multi-anisotropic Gevrey space Es,Γ (Ω), and then it introduces the space of multi-anisotropic ultradifferentiable functions E M,Γ (Ω) unifying these both spaces.
[1] A. Beurling. Quasi-analyticity and general distributions. Lecture 4 and 5, AMS Summer Institute, Standford, (1961).
[2] J. Bonet, R. Meise, S. N. Melikhov. A comparison of two different ways to define classes of ultradifferentiable functions. Bull. Belg. Math. Soc. Simon Stevin, Vol. 14-3, p. 425-444, (2007).
[3] C. Bouzar. Spaces of ultradifferentiable functions of multi-anisotropic type. In P. Boggiatto et al. (eds.), Advances in Microlocal and Time-Frequency, Springer Nature, p. 131-144, (2020).
[4] C. Bouzar, R. Chaïli. Vecteurs Gevrey de systèmes quasielliptiques. Annales de Mathématiques de l'Université de Sidi Bel Abbés, T. 5, p. 33-43, (1998).
