The Properties And Operations Of ω-Ultradistribution Space

The theories of partial differential equation have got remarkable development be-cause of the appearance of distribution theory. Since the sixties of last century, somemathematician have introduced conceptions of ultra-differentiable functions and ultra-distributions in order to satisfy the needs of different problems arised in the theory ofpartial differriential equations. These have expanded the theories of distributions.This article, based on the results of J.Bonet,R.Meise and B.A.Taylor, mainlystudies the properties of theω-ultradistributions, convolution operators and Fourier-Laplace transform on D′_* andε′_*. These will be helpful for us to further study thetheory of partial differential equations.In this paper, we obtain the following three results:Theorem A linear functional T∈D′_{ω}(Ω) if and only if for each compact K (?)Ωand eachε＞0, there exist C_K,ε＞0, such that: ||≤C_K,εintegral from n=(R^N)|(?)(t)|exp(εω(t))dt,f∈D_ω(K,ε).Theorem A linear functional T∈D′_ω(Ω) if and only if there existλ₀＞0, foreach compact K (?)Ωthere exist C_K＞0, such that for all,λ≥λ₀: ||≤C_K integral from n=(R^N)|(?)(t)|exp(λω(t))dt,f∈D_ω(Ω),supp f (?) KTheorem Let R, S, T∈D′_*(R^N)and at least two of them have compact support,then (1)(R*S)*T=R*(S*T); (2)R*T=T*R; (3)supp(R*T)(?)suppR+suppT; (4)δ*T=T*δ=δ.