ω-Ultradifferentiable Function Spaces And Operatinos

Since the fifties of last century, the theories of partial differential equation have got remarkable development because of the appearance of distribution theory. From the sixties, the notion of distribution has been extended in several way according to the needs of different problems and different fields.In the eighties of twenty's century, by modifying the condition of ultradifferential functions of Beurling , Retzsche and Vogt, R.Meise B.A.Taylor,D.Vogt and J.Bonet etc.introduced the ultradifferential functions of Beurling type ε(ω) (Ω) (resp of Roumieu type ε{w}(Ω)) and test fuctions of Beurling type D(ω)(Ω)(resp. of Roumieu type D{w}(Ω)). since then, many importent results have been obtained on the research of Fourier transform, convolution operator and linear partial different equations in these spaces.In this paper, we discuss some multiplication and convolution operations in ε(ω)(ε{ω}) and D(ω)(D{ω}) by Fourier -Laplace transform, and obtain the following results:Theorem 1 Let ω be a weight function, f∈D(R^N), g ∈ D*(R^N). Then we have:fg∈D*(R^N) and fg = (2π)^-nf*gTheorem 2 Let ω be a weight function, f ∈D(R^N), g ∈ ε*(R^N), Then we have: f*g∈ ε*(R^N).Theorem 3 If f, g∈D*(R^N),then we have:Where Z\, denote V^)(V^) and ￡* denote ￡(w...