Topological Isomorphism For ω-ultra-distribution Spaces

In the early1950s, L.Schwartz established strict basis for the theory of distributions, and set up the foundation for the study of partial differential equations in a broader class of functions. In the1960s, A.Beurling, G.Bjorck and H.Komatsu etc gave the concepts of ultra-differentiable functions and ultra-distributions using weight functions. To the1980s, Meise, Bonet and Taylor etc appropriately modified the condition of ultra-differentiable functions given by Beurling, Retzsche, Vogt, replaced subadditivity of the weight function by a weaker condition of a (see the definition of weight function ω), and introduced con-ceptions of ω-ultra-differentiable functions and w-ultra-distributions. Subsequently, many scholars studied intensively properties of ultra-distribution spaces D*’,ε*’,surjectivity and the existence of the right inverse of linear partial differential operators in these spaces, and many important results were obtained.Because of the importance of differential equations in the theory research and appli-cation of natural science, and because of the role of w-ultra-differentiable functions and w-ultra-distributions for the theory research of partial differential equations, the study on special characteristic and structure of w-ultra-differentiable function spaces and w-ultra-distribution spaces is still a hot pot in the present academic circle.In this thesis, we introduces four subspaces of entire function space H(CN):A(ω)(CN,Ω), A{ω}(CN,Ω), A(ω)(CN,Ω) and A{ω}(CN,Ω) for certain weight function w and open con-vex set Ω (?) RN. Then, we obtain isomorphisms between ω-ultra-differentiable function spaces D*, w-ultra-distribution spaces ε*’ and A(ω)(CN,Ω), A{ω}(CN,Ω), A(ω)(CN,Ω), A{ω}(CN,Ω) using Fourier-Laplace transform, so as to enable us to investigate these ω-ultra-distributions spaces by using the entire function space H(CN) that we have known.