The Problem Of Constructional Expressions For ω-Ultra-distribution Spaces

One of the most direct and important purposes of the partial differential equation theory is to research the existence of the equation solution. In the 1950s, the emergence of distributions provided a good tool for the research of weak solution to the equation with the nature of a leap, and produced theories of many mathematical branches. For example, the local analysis, the theory of quasi-differential operator, the theory of Fourier integral operator, super-functions, etc. Ultra-differentiable functions and ultra-distributions are important parts. In the 1960s, A.Beurling [1], G.Bjorck [2], and H.Komatsu [3-4] gave the concept of ultra-distributions using weight function. After the 80s, J.Bonet, R.W.Braun, R.Mise, B.A.Taylor and D.Vogt put it to ω-ultra-distributions again [5-10,13,17], make a thorough study on the characteristic of the spaces, and carry out on its right inverse existence of linear partial differential operator [11,12,14,15], obtain many important results.With the rapid development of technology and social economy, partial differential equations are widely used in the field of mathematics, physics, engineering, economy and so on. w-ultra-differentiable functions and ω-ultra-distributions are of great importance in the research and development of partial differential equations. Because of the complexity of the structures of w-ultra-differentiable function spaces and w-ultra-distribution spaces, the characteristic of their elements and the structures of these spaces are still a scientific phenomenon at present.Based on the above reasons, the spatial structures of w-ultra-differentiable functions and w-ultra-distributions are discussed by three chapters in this paper:The research background and current situation of ultra-differentiable functions and ultra-distributions are explained in the first chapter.The second chapter is mainly about basic concepts and basic properties. The concept of weight function ω is introduced, and the w-test function spaces D*, w-ultra-differentiable function spacesε* and w-ultra-distribution spaces D’* and S’* are defined by weight function ω. Then some basic properties are given.In the third chapter, for a open convex set Ω (?) RN, four subspaces of entire function space H(CN) are defined by weight function ω:A(ω)(CN,Ω), A(ω)(CN,Ω), A(ω)(CN,Ω) and A{ω}(CN,Ω). Then isomorphisms between w-test function space D*, ω-ultra-distribution space ε*’,D*’ and four subspaces of entire function space H(CN) are constructed by the Fourier-Laplace transform. Based on these, we obtain two spatial structure expressions about ε’* and D’*.