Structure Theorem for Functionals in the Space ? ω 1 , ω 2 ′

Abstract EN

We introduce the space ? ω 1 , ω 2 of all C ∞ functions ϕ such that sup | α | ≤ m ∥ e k ω 1 ∂ α ϕ ∥ ∞ and sup | α | ≤ m ∥ e k ω 2 ∂ α ϕ ^ ∥ ∞ are finite for all k ∈ ℕ 0 , α ∈ ℕ 0 n , where ω 1 and ω 2 are two weights satisfying the classical Beurling conditions.

Moreover, we give a topological characterization of the space ? ω 1 , ω 2 without conditions on the derivatives.

For functionals in the dual space ? ω 1 , ω 2 ′ , we prove a structure theorem by using the classical Riesz representation thoerem.