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Boundary Values and Convolution in Ultradistribution Spaces by Richard D. Carmichael

By Richard D. Carmichael

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For each ϕ ∈ B and k > 0 k | x β ϕ(α) (x)|2 dx ≤ 2 R\[−k,k] |x β+1 ϕ(α) (x)|2 dx R\[−k,k] ≤ Mα Mβ+1 . m ˜ α+β+1 Therefore | x β ϕ(α) (x)|2 dx ≤ k −2 Mα Mβ+1 , m ˜ α+β+1 ϕ ∈ B. R\[−k,k] According to the theorem of Kolmogorov, it follows that the set Bαβ , α, β ∈ N0 , is relatively compact in L2 . Let us prove that B fulfills condition (ii). For each ε > 0 there exists µ ∈ N0 such that mα ≤ εm ˜ α for all α ≥ µ. Hence α≥µ β∈N0 2 mα+β x β ϕ(α) (x) dx ≤ ε2 Mα Mβ R α≥µ β∈N0 2 m ˜ α+β x β ϕ(α) (x) dx Mα Mβ R ≤ ε2 for each ϕ ∈ B and the proof of assertion (ii) and the whole theorem is completed.

1. Proof of Part (2). We start by introducing additional notation for arbitrary t ∈ [1, ∞]. For a given function ψ denote by ψ Lt , ψ Lt and ψ Lt its Lt norms on the sets [−1, 1], R \ [−1, 1] and R, respectively. Let α, β, γ ∈ N0 and t ∈ [1, ∞). For a given smooth function ϕ denote aα,β (ϕ) := xβ ϕ(α) L∞ ; bα,β (ϕ) cα,β (ϕ) := xβ ϕ(α) Atα,β (ϕ) := xβ ϕ(α) t Lt ; Bα,β (ϕ) t Cα,β (ϕ) := xβ ϕ(α) := xβ ϕ(α) L∞ ; L∞ ; := xβ ϕ(α) Lt ; Lt . Moreover, denote by Iγ,t the Lt norm of the function τ (x) = x−γ on R \ [−1, 1] and R.

3 Functions and ultradistributions of Beurling and Roumieu type In this section, we introduce ultradifferentiable functions and various types of spaces of test functions and ultradistributions meant as elements of the respective dual spaces. May 14, 2007 15:29 World Scientific Book - 9in x 6in Ultradifferentiable Functions and Ultradistributions Newbook 21 Let (Mp ), p ∈ N0 , be a sequence of positive numbers. 32) for all h > 0 (respectively, for some h > 0). Here the positive constants N and h depend only on ϕ; they do not depend on α.

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