About Some M - M′ non Quasi-Analytic Classes

Abstract.

Let Ω and Ω′ be non empty open subsets of ${\mathbb{R}}^r$ and ${\mathbb{R}}^s$ respectively and let m and m′ be increasing and non quasi-analytic sequences of real numbers such that m ₀ = m′₀ = 1. We introduce the spaces $\varepsilon^{(M ,M^\prime)}(\Omega \times \Omega^\prime), {\mathcal{D}}^{(M, M^\prime)}(\Omega \times \Omega^\prime), \varepsilon^{\{M, M^\prime\}}(\Omega \times \Omega^\prime)$ and ${\mathcal{D}}^{\{M, M^\prime\}}(\Omega \times \Omega^\prime)$ without use of the Whitney extension theory. We study their locally convex properties, the structure of their elements and consider their links with the tensor products $\varepsilon ^*(\Omega) \bigotimes\varepsilon ^*(\Omega^\prime)$ and ${\mathcal{D}}^*(\Omega) \bigotimes {\mathcal{D}}^*(\Omega ^\prime)$ endowed with the ε-, π- and i-topologies. This leads to tensor product representations and kernel theorems.